24. ; Graph the function and observe where it is discontinuous. 2xy f x, y x 2 y 3 x 3 y xy 35. 6x 3 y 2x 4 y 4 36.

Size: px
Start display at page:

Download "24. ; Graph the function and observe where it is discontinuous. 2xy f x, y x 2 y 3 x 3 y xy 35. 6x 3 y 2x 4 y 4 36."

Transcription

1 SECTION. LIMITS AND CONTINUITY 877. EXERCISES. Suppose that, l 3, f, 6. What can ou sa about the value of f 3,? What if f is continuous?. Eplain wh each function is continuous or discontinuous. (a) The outdoor temperature as a function of longitude, latitude, and time (b) Elevation (height above sea level) as a function of longitude, latitude, and time (c) The cost of a tai ride as a function of distance traveled and time 3 Use a table of numerical values of f, for, near the origin to make a conjecture about the value of the it of f, as, l,. Then eplain wh our guess is correct. 3. f, f,. 5 6 Find h, t f, and the set on which h is continuous. 5. tt t st, 6. tt t ln t, ; 7 8 Graph the function and observe where it is discontinuous. Then use the formula to eplain what ou have observed. 7., l, 3 6 f, e f, 3 6 f, 8. f, 5 Find the it, if it eists, or show that the it does not eist , l, 7. 8., l, cos.., l, e 5. 6., l, 7. 8., l, s 9...., l,, l, e sin,, l 3,, 3,, l,,,, l,,,, l,, 3 s 9 ; 3 Use a computer graph of the function to eplain wh the it does not eist. 3 3., l, 3 5, l, e cos ln, l, sin, l,, l,, l,, l,, l, 6 3 sin Determine the set of points at which the function is continuous. 9. F, sin 3. F, e 3. F, arctan( s ) 3. F, e s 33. G, ln 3. G, tan ( ) f, Use polar coordinates to find the it. [If r, are polar coordinates of the point, with r, note that r l as, l,.] 39.. f,, f,, s f, 3 3, l, s ln, l, e., l, if,, if,, if,, if,,

2 Determine the signs of the partial derivatives for the function f whose graph is shown. SECTION.3 PARTIAL DERIVATIVES 889. A contour map is given for a function f. Use it to estimate f, and f, (a) f, (b) 6. (a) f, (b) 7. (a) f, (b) 8. (a) f, (b) f, f, f, f,. If f, 6, find f, and f, and interpret these numbers as slopes. Illustrate with either hand-drawn sketches or computer plots.. If f, s, find f, and f, and interpret these numbers as slopes. Illustrate with either handdrawn sketches or computer plots. 9. The following surfaces, labeled a, b, and c, are graphs of a function f and its partial derivatives f and f. Identif each surface and give reasons for our choices. ; 3 Find f and f and graph f, f, and f with domains and viewpoints that enable ou to see the relationships between them. 3.. f, e f, Find the first partial derivatives of the function. 5. f, f, f, t e t cos 8. f, t s ln t _ a _8 _3 3 _ f,.. 3. w sin cos. 5. f r, s r lnr s 6. tan f, w e v u v f, t arctan(st ) _ b _3 3 _ 7. u te wt f,, w ln u sin f,,, t tant u s n f, cost dt f,, sin w e u f,,, t t 8 8 _3 c 3 _ 38. u sin n n 39 Find the indicated partial derivatives. 39. f, ln( s ) ; f 3,. f, arctan; f, 3. f,, ; f,,

3 89 CHAPTER PARTIAL DERIVATIVES. f,, ssin sin sin ; f,, 69. Use the table of values of f, to estimate the values of f 3,, f 3,., and f 3,. 3 Use the definition of partial derivatives as its () to find f, and f,. 3. f, 3. f, Use implicit differentiation to find and ln 7. arctan 8. sin Find and. 9. (a) f t (b) f 5. (a) f t (b) f (c) f Level curves are shown for a function f. Determine whether the following partial derivatives are positive or negative at the point P. (a) f (b) f (c) f (d) f (e) f 8 6 P 5 56 Find all the second partial derivatives. 5. f, f, sin m n 53. w su v 5. v 55. arctan 56. v e e 57 6 Verif that the conclusion of Clairaut s Theorem holds, that is, u u. 57. u sin 58. u u ln s 6. u e 6 68 Find the indicated partial derivative. 6. f, 3 3 ;, 6. f, t e ct ;, 63. f,, cos 3 ;, 6. f r, s, t r lnrs t 3 ;, 65. u e r sin ; 66. usv w ; 3 w 67. w ;, 68. u a b c ; f ttt 3 u r 3 u v w 6 u 3 f t f f rss f f rst 3 w f f 7. Verif that the function u e k t sin k is a solution of the heat conduction equation u t u. 7. Determine whether each of the following functions is a solution of Laplace s equation u u. (a) u (b) u (c) u 3 3 (d) u ln s (e) u sin cosh cos sinh (f) u e cos e cos 73. Verif that the function u s is a solution of the three-dimensional Laplace equation u u u. 7. Show that each of the following functions is a solution of the wave equation u tt a u. (a) u sink sinakt (b) u ta t (c) u at 6 at 6 (d) u sin at ln at 75. If f and t are twice differentiable functions of a single variable, show that the function u, t f at t at is a solution of the wave equation given in Eercise If u e aa an n, where a a an, show that u u u n u 77. Verif that the function lne e is a solution of the differential equations

4 SECTION.3 PARTIAL DERIVATIVES 89 and 87. You are told that there is a function f whose partial derivatives are f, and f, 3. Should ou believe it? 78. Show that the Cobb-Douglas production function satisfies the equation 79. Show that the Cobb-Douglas production function satisfies PL, K C K L b solving the differential equation (See Equation 5.) 8. The temperature at a point, on a flat metal plate is given b T, 6, where T is measured in C and, in meters. Find the rate of change of temperature with respect to distance at the point, in (a) the -direction and (b) the -direction. Find RR. L P P K L K P 8. The gas law for a fied mass m of an ideal gas at absolute temperature T, pressure P, and volume V is PV mrt, where R is the gas constant. Show that 83. For the ideal gas of Eercise 8, show that P V dp dl P L R R R R 3 V T T P T 8. The wind-chill inde is modeled b the function W 3..65T.37v Tv.6 where T is the temperature C and v is the wind speed kmh. When T 5C and v 3 kmh, b how much would ou epect the apparent temperature W to drop if the actual temperature decreases b C? What if the wind speed increases b kmh? 85. The kinetic energ of a bod with mass m and velocit v is K mv. Show that K m T P V mr T K v K P blk 8. The total resistance R produced b three conductors with resistances R, R, R 3 connected in a parallel electrical circuit is given b the formula 86. If a, b, c are the sides of a triangle and A, B, C are the opposite angles, find Aa, Ab, Ac b implicit differentiation of the Law of Cosines. ; 88. The paraboloid 6 intersects the plane in a parabola. Find parametric equations for the tangent line to this parabola at the point,,. Use a computer to graph the paraboloid, the parabola, and the tangent line on the same screen. 89. The ellipsoid 6 intersects the plane in an ellipse. Find parametric equations for the tangent line to this ellipse at the point,,. 9. In a stud of frost penetration it was found that the temperature T at time t (measured in das) at a depth (measured in feet) can be modeled b the function where and is a positive constant. (a) Find T. What is its phsical significance? (b) Find Tt. What is its phsical significance? (c) Show that T satisfies the heat equation T t kt for a certain constant k. ; (d) If, T, and T, use a computer to graph T, t. (e) What is the phsical significance of the term in the epression sint? 9. Use Clairaut s Theorem to show that if the third-order partial derivatives of f are continuous, then 9. (a) How man nth-order partial derivatives does a function of two variables have? (b) If these partial derivatives are all continuous, how man of them can be distinct? (c) Answer the question in part (a) for a function of three variables. 93. If f, 3 e sin, find f,. [Hint: Instead of finding f, first, note that it s easier to use Equation or Equation.] 9. If f, s 3 3 3, find f,. 95. Let 365. T, t T T e sint 3 3 f, ; (a) Use a computer to graph f. (b) Find f, and f, when,,. (c) Find f, and f, using Equations and 3. (d) Show that f, and f,. CAS (e) Does the result of part (d) contradict Clairaut s Theorem? Use graphs of and to illustrate our answer. f f f f f if,, if,,

5 SECTION. TANGENT PLANES AND LINEAR APPROXIMATIONS 899. EXERCISES 6 Find an equation of the tangent plane to the given surface at the specified point.., , 3. s,. ln, 5. cos, 6. e, ; 7 8 Graph the surface and the tangent plane at the given point. (Choose the domain and viewpoint so that ou get a good view of both the surface and the tangent plane.) Then oom in until the surface and the tangent plane become indistinguishable. CAS 7. 3, 8. arctan, 9 Draw the graph of f and its tangent plane at the given point. (Use our computer algebra sstem both to compute the partial derivatives and to graph the surface and its tangent plane.) Then oom in until the surface and the tangent plane become indistinguishable. 9.. f, 6 Eplain wh the function is differentiable at the given point. Then find the lineariation L, of the function at that point.. f, s,,. f, 3,,,,, 3. f,,,,,. f, s e, 5. f, e cos,, 6. f, sin 3,,,,,,, 5,, sin,,, 3,, 3,,, f, e (s s s ),,, 3e. 7 8 Verif the linear approimation at, s cos 9. Find the linear approimation of the function f, s 7 at, and use it to approimate f.95,.8. ;. Find the linear approimation of the function f, ln 3 at 7, and use it to approimate f 6.9,.6. Illustrate b graphing f and the tangent plane.. Find the linear approimation of the function f,, s at 3,, 6 and use it to approimate the number s The wave heights h in the open sea depend on the speed v of the wind and the length of time t that the wind has been blowing at that speed. Values of the function h f v, t are recorded in feet in the following table. Wind speed (knots) Use the table to find a linear approimation to the wave height function when v is near knots and t is near hours. Then estimate the wave heights when the wind has been blowing for hours at 3 knots. 3. Use the table in Eample 3 to find a linear approimation to the heat inde function when the temperature is near 9F and the relative humidit is near 8%. Then estimate the heat inde when the temperature is 95F and the relative humidit is 78%.. The wind-chill inde W is the perceived temperature when the actual temperature is T and the wind speed is v, so we can write W f T, v. The following table of values is an ecerpt from Table in Section.. Actual temperature ( C) t v T v Wind speed (km/h) Duration (hours) Use the table to find a linear approimation to the wind-chill

6 9 CHAPTER PARTIAL DERIVATIVES inde function when T is near 5C and v is near 5 kmh. Then estimate the wind-chill inde when the temperature is 7C and the wind speed is 55 kmh. 5 3 Find the differential of the function ln 6. v cos v 7. m p 5 q 3 8. T uvw 9. R cos 3. w e 3. If 5 and, changes from, to.5,., compare the values of and d. 3. If 3 and, changes from 3, to.96,.95, compare the values of and d. 33. The length and width of a rectangle are measured as 3 cm and cm, respectivel, with an error in measurement of at most. cm in each. Use differentials to estimate the maimum error in the calculated area of the rectangle. 3. The dimensions of a closed rectangular bo are measured as 8 cm, 6 cm, and 5 cm, respectivel, with a possible error of. cm in each dimension. Use differentials to estimate the maimum error in calculating the surface area of the bo. 35. Use differentials to estimate the amount of tin in a closed tin can with diameter 8 cm and height cm if the tin is. cm thick. 36. Use differentials to estimate the amount of metal in a closed clindrical can that is cm high and cm in diameter if the metal in the top and bottom is. cm thick and the metal in the sides is.5 cm thick. 37. A boundar stripe 3 in. wide is painted around a rectangle whose dimensions are ft b ft. Use differentials to approimate the number of square feet of paint in the stripe. 38. The pressure, volume, and temperature of a mole of an ideal gas are related b the equation PV 8.3T, where P is measured in kilopascals, V in liters, and T in kelvins. Use differentials to find the approimate change in the pressure if the volume increases from L to.3 L and the temperature decreases from 3 K to 35 K. 39. If R is the total resistance of three resistors, connected in parallel, with resistances,,, then If the resistances are measured in ohms as R 5, R, and R 3 5, with a possible error of.5% in each case, estimate the maimum error in the calculated value of R.. Four positive numbers, each less than 5, are rounded to the first decimal place and then multiplied together. Use differentials to estimate the maimum possible error in the computed product that might result from the rounding.. A model for the surface area of a human bod is given b S.9w.5 h.75, where w is the weight (in pounds), h is the height (in inches), and S is measured in square feet. If the errors in measurement of w and h are at most %, use differentials to estimate the maimum percentage error in the calculated surface area.. Suppose ou need to know an equation of the tangent plane to a surface S at the point P,, 3. You don t have an equation for S but ou know that the curves both lie on S. Find an equation of the tangent plane at P. 3 Show that the function is differentiable b finding values of and that satisf Definition f, 5. Prove that if f is a function of two variables that is differentiable at a, b, then f is continuous at a, b. Hint: Show that 6. (a) The function r t 3t, t, 3 t t r u u, u 3, u f,. f a, b f a, b, l, was graphed in Figure. Show that f, and f, both eist but f is not differentiable at,. [Hint: Use the result of Eercise 5.] (b) Eplain wh and are not continuous at,. f R R R R R 3 f R R 3 if,, if f, 5,,

7 SECTION.5 THE CHAIN RULE 97 EXAMPLE 9 Find and if SOLUTION Let F,, Then, from Equations 7, we have N The solution to Eample 9 should be compared to the one in Eample in Section.3. F 3 6 F 3 6 F 3 6 F 3 6 M.5 EXERCISES 6 Use the Chain Rule to find ddt or dwdt.., sin t, e t. cos, 5t, t 3. s, ln t, cos t. tan, e t, e t 5. w e, t, t, t 6. w lns, sin t, cos t, 7 Use the Chain Rule to find s and t. 7. 3, s cos t, s sin t 8. arcsin, s t, st st 9. sin cos,, s t tan t. Let Ws, t Fus, t, vs, t, where F, u, and v are differentiable, and u, v, 3 u s, v s, 5 u t, 6 v t, F u, 3 F v, 3 Find W s, and W t,. 5. Suppose f is a differentiable function of and, and tu, v f e u sin v, e u cos v. Use the table of values to calculate t u, and t v,. f t f f, 3 6 8, e, st,. e r cos, r st, ts ss t 6. Suppose f is a differentiable function of and, and tr, s f r s, s r. Use the table of values in Eercise 5 to calculate t r, and t s,.. tanuv, u s 3t, 3. If f,, where f is differentiable, and tt t3 t3 5 f, 7 6 find ddt when t 3. v 3s t ht h3 7 h3 f, Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable. 7. u f,, where r, s, t, r, s, t 8. R f,,, t, where u, v, w, u, v, w, u, v, w, t tu, v, w 9. w f r, s, t, where r r,, s s,, t t,. t f u, v, w, where u up, q, r, s, v vp, q, r, s, w wp, q, r, s

8 98 CHAPTER PARTIAL DERIVATIVES 6 Use the Chain Rule to find the indicated partial derivatives.. 3, uv w 3, u ve w ;,, when u, v, w u v w. u sr s, r cos t, s sin t; u u u,, when,, t t 3. R lnu v w, u, v, w ; R R, when. M e, uv, u v, u v; M M, when u 3, v u v 5. u, pr cos, pr sin, p r; u u u,, when p, r 3, p r 6. Y w tan uv, u r s, v s t, w t r; Y Y Y,, when r, s, t r s t 7 3 Use Equation 6 to find dd. 7. s e 9. cos e 3. sin cos sin cos 3 3 Use Equations 7 to find and cos 33. arctan 3. ln 35. The temperature at a point, is T,, measured in degrees Celsius. A bug crawls so that its position after t seconds is given b s t, 3 t, where and are measured in centimeters. The temperature function satisfies T, 3 and T, 3 3. How fast is the temperature rising on the bug s path after 3 seconds? 36. Wheat production W in a given ear depends on the average temperature T and the annual rainfall R. Scientists estimate that the average temperature is rising at a rate of.5 Cear and rainfall is decreasing at a rate of. cmear. The also estimate that, at current production levels, WT and WR 8. (a) What is the significance of the signs of these partial derivatives? (b) Estimate the current rate of change of wheat production, dwdt. 37. The speed of sound traveling through ocean water with salinit 35 parts per thousand has been modeled b the equation where C is the speed of sound (in meters per second), T is the temperature (in degrees Celsius), and D is the depth below the ocean surface (in meters). A scuba diver began a leisurel dive into the ocean water; the diver s depth and the surrounding water temperature over time are recorded in the following graphs. Estimate the rate of change (with respect to time) of the speed of sound through the ocean water eperienced b the diver minutes into the dive. What are the units? D 5 5 C 9..6T.55T.9T 3.6D 3 t (min) 3 t (min) 38. The radius of a right circular cone is increasing at a rate of.8 ins while its height is decreasing at a rate of.5 ins. At what rate is the volume of the cone changing when the radius is in. and the height is in.? 39. The length, width w, and height h of a bo change with time. At a certain instant the dimensions are m and w h m, and and w are increasing at a rate of ms while h is decreasing at a rate of 3 ms. At that instant find the rates at which the following quantities are changing. (a) The volume (b) The surface area (c) The length of a diagonal. The voltage V in a simple electrical circuit is slowl decreasing as the batter wears out. The resistance R is slowl increasing as the resistor heats up. Use Ohm s Law, V IR, to find how the current I is changing at the moment when R, I.8 A, dvdt. Vs, and drdt.3 s.. The pressure of mole of an ideal gas is increasing at a rate of.5 kpas and the temperature is increasing at a rate of.5 Ks. Use the equation in Eample to find the rate of change of the volume when the pressure is kpa and the temperature is 3 K.. Car A is traveling north on Highwa 6 and car B is traveling west on Highwa 83. Each car is approaching the intersection of these highwas. At a certain moment, car A is.3 km from the intersection and traveling at 9 kmh while car B is. km from the intersection and traveling at 8 kmh. How fast is the distance between the cars changing at that moment? 3. One side of a triangle is increasing at a rate of 3 cms and a second side is decreasing at a rate of cms. If the area of the T 6 8

9 APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES A9 EXERCISES. PAGE 877 N. Nothing; if f is continuous, f 3, Does not eist. Does not eist Does not eist Does not eist 3. The graph shows that the function approaches different numbers along different lines. 5. h, 3 6 s 3 6;, Along the line 9., e 3., 33., 35.,,, s 37.,,, f, f _ f _ f _ 3. f is continuous on EXERCISES.3 N PAGE 888. (a) The rate of change of temperature as longitude varies, with latitude and time fied; the rate of change as onl latitude varies; the rate of change as onl time varies. (b) Positive, negative, positive 3. (a) f T5, 3.3; for a temperature of 5C and wind speed of 3 kmh, the wind-chill inde rises b.3c for each degree the temperature increases. f v5, 3.5; for a temperature of 5C and wind speed of 3 kmh, the wind-chill inde decreases b.5c for each kmh the wind speed increases. (b) Positive, negative (c) 5. (a) Positive (b) Negative 7. (a) Positive (b) Negative 9. c f, b f, a f. f, 8 slope of C, f, slope of C 6 _ (, ) (,, 8) C 6 C (, ) (,, 8) 5. f, 3, f, f, t e t sin, f t, t e t cos , f,, f, f, f sr, s rs rr, s r r s lnr s r s 7. ut e wt ( wt), uw e wt 9. f, f, f w 3, w 3, w u sin, u sin s, u s w cos cos, w sin sin 35. f, f t sec t tant tant, f, f t sec tant t u i is n f, 3, f, , , 9. (a) f, t (b) f, f 5. f, f, f f 53. w, w uv uvu v 3 uu v u v 3 w vu, w vv u u v 3 55.,, f

10 A APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES 6., , e r sin 3, sin 3 u u u u u u ,, sin cos r sin, r r r s s s R., 6.8, 3.5 R u u u 87. No 89. t,, t t t t 93. w w r w s 9. w t, 95. (a) r s t w w r w s w t. r s t 9. 85, 78, , ,, 3 _. 3 sin e _ _ s sin e , (b), f, 5 3 f, , (c), (e) No, since f and f are not continuous. 35. Cs ms per minute 39. (a) 6 m 3 s (b) m s (c) ms EXERCISES. N PAGE Ls 3. (s3) rads (a) r cos sin, rs r s rs r sin r cos _ ; ; T H 39; 9F d 3 ln d 3 d dm 5p q 3 dp 3p 5 q dq dr cos d cos d sin d 3..95, d cm cm % 3., EXERCISES.5 N PAGE 97. cos t e t 3. t sin ts 5. e t 7. s 3 cos t 3 sin t, t s 3 sin t 3s cos t 9. s t cos cos st sin sin, t st cos cos s sin sin., s e rt s cos ss t t e rs t cos sin ss t sin 5 _5 _ EXERCISES.6 N PAGE 9..8 mbkm s3 7. (a) f, cos 3, 3 cos 3 (b), 3 (c) s (a) e, e, e (b),, (c) s 5. s3 7. 9(s5) s,, 3.,, 5., 3, 6, 7. (b), 9 9. All points on the line 3. (a) (3s3) (a) 3s3 (b) 38, 6, (c) s (a) (b) (a) (b) (a) (b) 5. 7., 3, 3 _ +3= 53. No 59. t, 6t, t 63. If u a, b and v c, d, then af bf and cf df are known, so we solve linear equations for and. f =6 (3, ) f Î f (3, )

Wind speed (knots) 8. Find and sketch the domain of the function. 9. Let f x, y, z e sz x 2 y Let t x, y, z ln 25 x 2 y 2 z 2.

Wind speed (knots) 8. Find and sketch the domain of the function. 9. Let f x, y, z e sz x 2 y Let t x, y, z ln 25 x 2 y 2 z 2. . 866 CHAPTER PARTIAL DERIVATIVES. The temperature-humidit inde I (or humide, for short) is the perceived air temperature when the actual temperature is T and the relative humidit is h, so we can write

More information

14.5 The Chain Rule. dx dt

14.5 The Chain Rule. dx dt SECTION 14.5 THE CHAIN RULE 931 27. w lns 2 2 z 2 28. w e z 37. If R is the total resistance of three resistors, connected in parallel, with resistances,,, then R 1 R 2 R 3 29. If z 5 2 2 and, changes

More information

Derivatives of Multivariable Functions

Derivatives of Multivariable Functions Chapter 0 Derivatives of Multivariable Functions 0. Limits Motivating Questions In this section, we strive to understand the ideas generated b the following important questions: What do we mean b the limit

More information

CHAPTER 3 Applications of Differentiation

CHAPTER 3 Applications of Differentiation CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. 0 Section. Rolle s Theorem and the Mean Value Theorem. 07 Section. Increasing and Decreasing Functions and the First

More information

v t t t t a t v t d dt t t t t t 23.61

v t t t t a t v t d dt t t t t t 23.61 SECTION 4. MAXIMUM AND MINIMUM VALUES 285 The values of f at the endpoints are f 0 0 and f 2 2 6.28 Comparing these four numbers and using the Closed Interval Method, we see that the absolute minimum value

More information

13.1. For further details concerning the physics involved and animations of the trajectories of the particles, see the following websites:

13.1. For further details concerning the physics involved and animations of the trajectories of the particles, see the following websites: 8 CHAPTER VECTOR FUNCTIONS N Some computer algebra sstems provide us with a clearer picture of a space curve b enclosing it in a tube. Such a plot enables us to see whether one part of a curve passes in

More information

18 19 Find the extreme values of f on the region described by the inequality. 20. Consider the problem of maximizing the function

18 19 Find the extreme values of f on the region described by the inequality. 20. Consider the problem of maximizing the function 940 CHAPTER 14 PARTIAL DERIVATIVES 14.8 EXERCISES 1. Pictured are a contour map of f and a curve with equation t, y 8. Estimate the maimum and minimum values of f subject to the constraint that t, y 8.

More information

Math 53 Homework 4 Solutions

Math 53 Homework 4 Solutions Math 5 Homework 4 Solutions Problem 1. (a) z = is a paraboloid with its highest point at (0,0,) and intersecting the -plane at the circle + = of radius. (or: rotate the parabola z = in the z-plane about

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Eam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The graph of a function is given. Choose the answer that represents the graph of its derivative.

More information

CHAPTER 3 Applications of Differentiation

CHAPTER 3 Applications of Differentiation CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. Section. Rolle s Theorem and the Mean Value Theorem. 7 Section. Increasing and Decreasing Functions and the First Derivative

More information

CHAPTER 3 Applications of Differentiation

CHAPTER 3 Applications of Differentiation CHAPTER Applications of Differentiation Section. Etrema on an Interval................... 0 Section. Rolle s Theorem and the Mean Value Theorem...... 0 Section. Increasing and Decreasing Functions and

More information

14.1 FUNCTIONS OF SEVERAL VARIABLES. In this section we study functions of two or more variables from four points of view:

14.1 FUNCTIONS OF SEVERAL VARIABLES. In this section we study functions of two or more variables from four points of view: 4 PARTIAL DERIVATIVES Functions of two variables can be visualied b means of level curves, which connect points where the function takes on a given value. Atmospheric pressure at a given time is a function

More information

MA261-A Calculus III 2006 Fall Homework 7 Solutions Due 10/20/2006 8:00AM

MA261-A Calculus III 2006 Fall Homework 7 Solutions Due 10/20/2006 8:00AM MA26-A Calculus III 2006 Fall Homework 7 Solutions Due 0/20/2006 8:00AM 3 #4 Find the rst partial derivatives of the function f (; ) 5 + 3 3 2 + 3 4 f (; ) 5 4 + 9 2 2 + 3 4 f (; ) 6 3 + 2 3 3 #6 Find

More information

UNIVERSITI TEKNOLOGI MALAYSIA SSE 1893 ENGINEERING MATHEMATICS TUTORIAL Determine the domain and range for each of the following functions.

UNIVERSITI TEKNOLOGI MALAYSIA SSE 1893 ENGINEERING MATHEMATICS TUTORIAL Determine the domain and range for each of the following functions. UNIVERSITI TEKNOLOGI MALAYSIA SSE 1893 ENGINEERING MATHEMATICS TUTORIAL 1 1 Determine the domain and range for each of the following functions a = + b = 1 c = d = ln( ) + e = e /( 1) Sketch the level curves

More information

AP Calc AB First Semester Review

AP Calc AB First Semester Review AP Calc AB First Semester Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the limit. 1) lim (7-7) 7 A) -4 B) -56 C) 4 D) 56 1) Determine

More information

Derivatives of Multivariable Functions

Derivatives of Multivariable Functions Chapter 10 Derivatives of Multivariable Functions 10.1 Limits Motivating Questions What do we mean b the limit of a function f of two variables at a point (a, b)? What techniques can we use to show that

More information

Mat 267 Engineering Calculus III Updated on 9/19/2010

Mat 267 Engineering Calculus III Updated on 9/19/2010 Chapter 11 Partial Derivatives Section 11.1 Functions o Several Variables Deinition: A unction o two variables is a rule that assigns to each ordered pair o real numbers (, ) in a set D a unique real number

More information

Answers to Some Sample Problems

Answers to Some Sample Problems Answers to Some Sample Problems. Use rules of differentiation to evaluate the derivatives of the following functions of : cos( 3 ) ln(5 7 sin(3)) 3 5 +9 8 3 e 3 h 3 e i sin( 3 )3 +[ ln ] cos( 3 ) [ln(5)

More information

y sin n x dx cos x sin n 1 x n 1 y sin n 2 x cos 2 x dx y sin n xdx cos x sin n 1 x n 1 y sin n 2 x dx n 1 y sin n x dx

y sin n x dx cos x sin n 1 x n 1 y sin n 2 x cos 2 x dx y sin n xdx cos x sin n 1 x n 1 y sin n 2 x dx n 1 y sin n x dx SECTION 7. INTEGRATION BY PARTS 57 EXAPLE 6 Prove the reduction formula N Equation 7 is called a reduction formula because the eponent n has been reduced to n and n. 7 sin n n cos sinn n n sin n where

More information

Review Test 2. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. D) ds dt = 4t3 sec 2 t -

Review Test 2. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. D) ds dt = 4t3 sec 2 t - Review Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the derivative. ) = 7 + 0 sec ) A) = - 7 + 0 tan B) = - 7-0 csc C) = 7-0 sec tan

More information

CHAPTER 6 Applications of Integration

CHAPTER 6 Applications of Integration PART II CHAPTER Applications of Integration Section. Area of a Region Between Two Curves.......... Section. Volume: The Disk Method................. 7 Section. Volume: The Shell Method................

More information

Math 323 Exam 2 - Practice Problem Solutions. 2. Given the vectors a = 1,2,0, b = 1,0,2, and c = 0,1,1, compute the following:

Math 323 Exam 2 - Practice Problem Solutions. 2. Given the vectors a = 1,2,0, b = 1,0,2, and c = 0,1,1, compute the following: Math 323 Eam 2 - Practice Problem Solutions 1. Given the vectors a = 2,, 1, b = 3, 2,4, and c = 1, 4,, compute the following: (a) A unit vector in the direction of c. u = c c = 1, 4, 1 4 =,, 1+16+ 17 17

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) = 2t + 1; D) = 2 - t;

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) = 2t + 1; D) = 2 - t; Eam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Calculate the derivative of the function. Then find the value of the derivative as specified.

More information

MATH 150/GRACEY PRACTICE FINAL. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MATH 150/GRACEY PRACTICE FINAL. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. MATH 0/GRACEY PRACTICE FINAL Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Choose the graph that represents the given function without using

More information

13.1. For further details concerning the physics involved and animations of the trajectories of the particles, see the following websites:

13.1. For further details concerning the physics involved and animations of the trajectories of the particles, see the following websites: 8 CHAPTER 3 VECTOR FUNCTIONS N Some computer algebra sstems provie us with a clearer picture of a space curve b enclosing it in a tube. Such a plot enables us to see whether one part of a curve passes

More information

CHAPTER 2: Partial Derivatives. 2.2 Increments and Differential

CHAPTER 2: Partial Derivatives. 2.2 Increments and Differential CHAPTER : Partial Derivatives.1 Definition of a Partial Derivative. Increments and Differential.3 Chain Rules.4 Local Etrema.5 Absolute Etrema 1 Chapter : Partial Derivatives.1 Definition of a Partial

More information

D u f f x h f y k. Applying this theorem a second time, we have. f xx h f yx k h f xy h f yy k k. f xx h 2 2 f xy hk f yy k 2

D u f f x h f y k. Applying this theorem a second time, we have. f xx h f yx k h f xy h f yy k k. f xx h 2 2 f xy hk f yy k 2 93 CHAPTER 4 PARTIAL DERIVATIVES We close this section b giving a proof of the first part of the Second Derivatives Test. Part (b) has a similar proof. PROOF OF THEOREM 3, PART (A) We compute the second-order

More information

11.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR

11.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR SECTION 11.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR 633 wit speed v o along te same line from te opposite direction toward te source, ten te frequenc of te sound eard b te observer is were c is

More information

Functions of Several Variables

Functions of Several Variables Chapter 1 Functions of Several Variables 1.1 Introduction A real valued function of n variables is a function f : R, where the domain is a subset of R n. So: for each ( 1,,..., n ) in, the value of f is

More information

Tangent Line Approximations

Tangent Line Approximations 60_009.qd //0 :8 PM Page SECTION.9 Dierentials Section.9 EXPLORATION Tangent Line Approimation Use a graphing utilit to graph. In the same viewing window, graph the tangent line to the graph o at the point,.

More information

CHAPTER 11 Vector-Valued Functions

CHAPTER 11 Vector-Valued Functions CHAPTER Vector-Valued Functions Section. Vector-Valued Functions...................... 9 Section. Differentiation and Integration of Vector-Valued Functions.... Section. Velocit and Acceleration.....................

More information

f x, y x 2 y 2 2x 6y 14. Then

f x, y x 2 y 2 2x 6y 14. Then SECTION 11.7 MAXIMUM AND MINIMUM VALUES 645 absolute minimum FIGURE 1 local maimum local minimum absolute maimum Look at the hills and valles in the graph of f shown in Figure 1. There are two points a,

More information

2.5 CONTINUITY. a x. Notice that Definition l implicitly requires three things if f is continuous at a:

2.5 CONTINUITY. a x. Notice that Definition l implicitly requires three things if f is continuous at a: SECTION.5 CONTINUITY 9.5 CONTINUITY We noticed in Section.3 that the it of a function as approaches a can often be found simpl b calculating the value of the function at a. Functions with this propert

More information

Calculus-Lab ) f(x) = 1 x. 3.8) f(x) = arcsin( x+1., prove the equality cosh 2 x sinh 2 x = 1. Calculus-Lab ) lim. 2.7) lim. 2.

Calculus-Lab ) f(x) = 1 x. 3.8) f(x) = arcsin( x+1., prove the equality cosh 2 x sinh 2 x = 1. Calculus-Lab ) lim. 2.7) lim. 2. ) Solve the following inequalities.) ++.) 4 > 3.3) Calculus-Lab { + > + 5 + < 3 +. ) Graph the functions f() = 3, g() = + +, h() = 3 cos( ), r() = 3 +. 3) Find the domain of the following functions 3.)

More information

Math 1431 Final Exam Review. 1. Find the following limits (if they exist): lim. lim. lim. lim. sin. lim. cos. lim. lim. lim. n n.

Math 1431 Final Exam Review. 1. Find the following limits (if they exist): lim. lim. lim. lim. sin. lim. cos. lim. lim. lim. n n. . Find the following its (if they eist: sin 7 a. 0 9 5 b. 0 tan( 8 c. 4 d. e. f. sin h0 h h cos h0 h h Math 4 Final Eam Review g. h. i. j. k. cos 0 n nn e 0 n arctan( 0 4 l. 0 sin(4 m. cot 0 = n. = o.

More information

UNIVERSITI TEKNOLOGI MALAYSIA SSE 1893 ENGINEERING MATHEMATICS TUTORIAL Determine the domain and range for each of the following functions.

UNIVERSITI TEKNOLOGI MALAYSIA SSE 1893 ENGINEERING MATHEMATICS TUTORIAL Determine the domain and range for each of the following functions. UNIVERSITI TEKNOLOGI MALAYSIA SSE 1893 ENGINEERING MATHEMATICS TUTORIAL 1 1 Determine the domain and range for each of the following functions a = + b = 1 c = d = ln( ) + e = e /( 1) Sketch the level curves

More information

10.3 Solving Nonlinear Systems of Equations

10.3 Solving Nonlinear Systems of Equations 60 CHAPTER 0 Conic Sections Identif whether each equation, when graphed, will be a parabola, circle, ellipse, or hperbola. Then graph each equation.. - 7 + - =. = +. = + + 6. + 9 =. 9-9 = 6. 6 - = 7. 6

More information

4.317 d 4 y. 4 dx d 2 y dy. 20. dt d 2 x. 21. y 3y 3y y y 6y 12y 8y y (4) y y y (4) 2y y 0. d 4 y 26.

4.317 d 4 y. 4 dx d 2 y dy. 20. dt d 2 x. 21. y 3y 3y y y 6y 12y 8y y (4) y y y (4) 2y y 0. d 4 y 26. 38 CHAPTER 4 HIGHER-ORDER DIFFERENTIAL EQUATIONS sstems are also able, b means of their dsolve commands, to provide eplicit solutions of homogeneous linear constant-coefficient differential equations.

More information

Additional Topics in Differential Equations

Additional Topics in Differential Equations 0537_cop6.qd 0/8/08 :6 PM Page 3 6 Additional Topics in Differential Equations In Chapter 6, ou studied differential equations. In this chapter, ou will learn additional techniques for solving differential

More information

2. Find the value of y for which the line through A and B has the given slope m: A(-2, 3), B(4, y), 2 3

2. Find the value of y for which the line through A and B has the given slope m: A(-2, 3), B(4, y), 2 3 . Find an equation for the line that contains the points (, -) and (6, 9).. Find the value of y for which the line through A and B has the given slope m: A(-, ), B(4, y), m.. Find an equation for the line

More information

MATH 150/GRACEY EXAM 2 PRACTICE/CHAPTER 2. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MATH 150/GRACEY EXAM 2 PRACTICE/CHAPTER 2. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. MATH 0/GRACEY EXAM PRACTICE/CHAPTER Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the indicated derivative. ) Find if = 8 sin. A) = 8

More information

y sec 3 x dx sec x tan x y sec x tan 2 x dx y sec 3 x dx 1 2(sec x tan x ln sec x tan x ) C

y sec 3 x dx sec x tan x y sec x tan 2 x dx y sec 3 x dx 1 2(sec x tan x ln sec x tan x ) C SECTION 7. TRIGONOETRIC INTEGRLS 465 Then sec 3 sec tan sec tan sec tan sec sec sec tan sec 3 sec Using ormula and solving for the required integral, we get sec 3 (sec tan ln sec tan ) C Integrals such

More information

6.7 Variation and Problem Solving. OBJECTIVES 1 Solve Problems Involving Direct Variation. 2 Solve Problems Involving Inverse Variation.

6.7 Variation and Problem Solving. OBJECTIVES 1 Solve Problems Involving Direct Variation. 2 Solve Problems Involving Inverse Variation. 390 CHAPTER 6 Rational Epressions 66. A doctor recorded a body-mass inde of 7 on a patient s chart. Later, a nurse notices that the doctor recorded the patient s weight as 0 pounds but neglected to record

More information

6 = 1 2. The right endpoints of the subintervals are then 2 5, 3, 7 2, 4, 2 9, 5, while the left endpoints are 2, 5 2, 3, 7 2, 4, 9 2.

6 = 1 2. The right endpoints of the subintervals are then 2 5, 3, 7 2, 4, 2 9, 5, while the left endpoints are 2, 5 2, 3, 7 2, 4, 9 2. 5 THE ITEGRAL 5. Approimating and Computing Area Preliminar Questions. What are the right and left endpoints if [, 5] is divided into si subintervals? If the interval [, 5] is divided into si subintervals,

More information

Review Exercises for Chapter 2

Review Exercises for Chapter 2 Review Eercises for Chapter 367 Review Eercises for Chapter. f 1 1 f f f lim lim 1 1 1 1 lim 1 1 1 1 lim 1 1 lim lim 1 1 1 1 1 1 1 1 1 4. 8. f f f f lim lim lim lim lim f 4, 1 4, if < if (a) Nonremovable

More information

Answer Key 1973 BC 1969 BC 24. A 14. A 24. C 25. A 26. C 27. C 28. D 29. C 30. D 31. C 13. C 12. D 12. E 3. A 32. B 27. E 34. C 14. D 25. B 26.

Answer Key 1973 BC 1969 BC 24. A 14. A 24. C 25. A 26. C 27. C 28. D 29. C 30. D 31. C 13. C 12. D 12. E 3. A 32. B 27. E 34. C 14. D 25. B 26. Answer Key 969 BC 97 BC. C. E. B. D 5. E 6. B 7. D 8. C 9. D. A. B. E. C. D 5. B 6. B 7. B 8. E 9. C. A. B. E. D. C 5. A 6. C 7. C 8. D 9. C. D. C. B. A. D 5. A 6. B 7. D 8. A 9. D. E. D. B. E. E 5. E.

More information

(x a) (a, b, c) P. (z c) E (y b)

(x a) (a, b, c) P. (z c) E (y b) ( a). FUNCTIONS OF TWO VARIABLES 67 G (,, ) ( c) (a, b, c) P E ( b) Figure.: The diagonal PGgives the distance between the points (,, ) and (a, b, c) F Using Pthagoras theorem twice gives (PG) =(PF) +(FG)

More information

MA 114 Worksheet #01: Integration by parts

MA 114 Worksheet #01: Integration by parts Fall 8 MA 4 Worksheet Thursday, 3 August 8 MA 4 Worksheet #: Integration by parts. For each of the following integrals, determine if it is best evaluated by integration by parts or by substitution. If

More information

Conic Sections CHAPTER OUTLINE. The Circle Ellipses and Hyperbolas Second-Degree Inequalities and Nonlinear Systems FIGURE 1

Conic Sections CHAPTER OUTLINE. The Circle Ellipses and Hyperbolas Second-Degree Inequalities and Nonlinear Systems FIGURE 1 088_0_p676-7 /7/0 :5 PM Page 676 (FPG International / Telegraph Colour Librar) Conic Sections CHAPTER OUTLINE. The Circle. Ellipses and Hperbolas.3 Second-Degree Inequalities and Nonlinear Sstems O ne

More information

Problems to practice for FINAL. 1. Below is the graph of a function ( ) At which of the marked values ( and ) is: (a) ( ) greatest = (b) ( ) least

Problems to practice for FINAL. 1. Below is the graph of a function ( ) At which of the marked values ( and ) is: (a) ( ) greatest = (b) ( ) least Problems to practice for FINAL. Below is the graph of a function () At which of the marked values ( and ) is: (a) () greatest = (b) () least = (c) () the greatest = (d) () the least = (e) () = = (f) ()

More information

CHAPTER 3 Applications of Differentiation

CHAPTER 3 Applications of Differentiation CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. 78 Section. Rolle s Theorem and the Mean Value Theorem. 8 Section. Increasing and Decreasing Functions and the First

More information

(2) Find the domain of f (x) = 2x3 5 x 2 + x 6

(2) Find the domain of f (x) = 2x3 5 x 2 + x 6 CHAPTER FUNCTIONS AND MODELS () Determine whether the curve is the graph of a function of. If it is state the domain and the range of the function. 5 8 Determine whether the curve is the graph of a function

More information

Find the slope of the curve at the given point P and an equation of the tangent line at P. 1) y = x2 + 11x - 15, P(1, -3)

Find the slope of the curve at the given point P and an equation of the tangent line at P. 1) y = x2 + 11x - 15, P(1, -3) Final Exam Review AP Calculus AB Find the slope of the curve at the given point P and an equation of the tangent line at P. 1) y = x2 + 11x - 15, P(1, -3) Use the graph to evaluate the limit. 2) lim x

More information

Math 1A Test Fall 10 Show your work for credit. Write all responses on separate paper. Don t use a calculator.

Math 1A Test Fall 10 Show your work for credit. Write all responses on separate paper. Don t use a calculator. Math A - 3.7 Test Fall 0 Name Show your work for credit. Write all responses on separate paper. Don t use a calculator.. Open cylindrical tubes resonate at the approimate frequencies, f, where nv f = L+

More information

PART A: Answer in the space provided. Each correct answer is worth one mark each.

PART A: Answer in the space provided. Each correct answer is worth one mark each. PART A: Answer in the space provided. Each correct answer is worth one mark each. 1. Find the slope of the tangent to the curve at the point (,6). =. If the tangent line to the curve k( ) = is horizontal,

More information

(b) Find the difference quotient. Interpret your result. 3. Find the average rate of change of ƒ(x) = x 2-3x from

(b) Find the difference quotient. Interpret your result. 3. Find the average rate of change of ƒ(x) = x 2-3x from 6360_ch0pp00-075.qd 0/6/08 4:8 PM Page 67 CHAPTER Summar 67 69. ƒ() = 3 70. ƒ() = -5 (b) Find the difference quotient. Interpret our result. 7. ƒ() = - 7. ƒ() = 0 73. ƒ() = + 74. ƒ() = -3 + 4 75. ƒ() =

More information

MATHEMATICS 200 December 2013 Final Exam Solutions

MATHEMATICS 200 December 2013 Final Exam Solutions MATHEMATICS 2 December 21 Final Eam Solutions 1. Short Answer Problems. Show our work. Not all questions are of equal difficult. Simplif our answers as much as possible in this question. (a) The line L

More information

CHAPTER 5 Logarithmic, Exponential, and Other Transcendental Functions

CHAPTER 5 Logarithmic, Exponential, and Other Transcendental Functions CHAPTER 5 Logarithmic, Eponential, and Other Transcendental Functions Section 5. The Natural Logarithmic Function: Differentiation.... 9 Section 5. The Natural Logarithmic Function: Integration...... 98

More information

6. Graph each of the following functions. What do you notice? What happens when x = 2 on the graph of b?

6. Graph each of the following functions. What do you notice? What happens when x = 2 on the graph of b? Pre Calculus Worksheet 1. Da 1 1. The relation described b the set of points {(-,5,0,5,,8,,9 ) ( ) ( ) ( )} is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph

More information

TO THE STUDENT: To best prepare for Test 4, do all the problems on separate paper. The answers are given at the end of the review sheet.

TO THE STUDENT: To best prepare for Test 4, do all the problems on separate paper. The answers are given at the end of the review sheet. MATH TEST 4 REVIEW TO THE STUDENT: To best prepare for Test 4, do all the problems on separate paper. The answers are given at the end of the review sheet. PART NON-CALCULATOR DIRECTIONS: The problems

More information

Appendix D: Variation

Appendix D: Variation A96 Appendi D Variation Appendi D: Variation Direct Variation There are two basic types of linear models. The more general model has a y-intercept that is nonzero. y m b, b 0 The simpler model y k has

More information

Increasing and Decreasing Functions and the First Derivative Test. Increasing and Decreasing Functions. Video

Increasing and Decreasing Functions and the First Derivative Test. Increasing and Decreasing Functions. Video SECTION and Decreasing Functions and the First Derivative Test 79 Section and Decreasing Functions and the First Derivative Test Determine intervals on which a unction is increasing or decreasing Appl

More information

Full file at

Full file at . Find the equation of the tangent line to y 6 at. y 9 y y 9 y Ans: A Difficulty: Moderate Section:.. Find an equation of the tangent line to y = f() at =. f y = 6 + 8 y = y = 6 + 8 y = + Ans: D Difficulty:

More information

One of the most common applications of Calculus involves determining maximum or minimum values.

One of the most common applications of Calculus involves determining maximum or minimum values. 8 LESSON 5- MAX/MIN APPLICATIONS (OPTIMIZATION) One of the most common applications of Calculus involves determining maimum or minimum values. Procedure:. Choose variables and/or draw a labeled figure..

More information

Graph and Write Equations of Ellipses. You graphed and wrote equations of parabolas and circles. You will graph and write equations of ellipses.

Graph and Write Equations of Ellipses. You graphed and wrote equations of parabolas and circles. You will graph and write equations of ellipses. TEKS 9.4 a.5, A.5.B, A.5.C Before Now Graph and Write Equations of Ellipses You graphed and wrote equations of parabolas and circles. You will graph and write equations of ellipses. Wh? So ou can model

More information

Bryn Mawr College Department of Physics Mathematics Readiness Examination for Introductory Physics

Bryn Mawr College Department of Physics Mathematics Readiness Examination for Introductory Physics Brn Mawr College Department of Phsics Mathematics Readiness Eamination for Introductor Phsics There are 7 questions and ou should do this eam in two and a half hours. Do not use an books, calculators,

More information

1. By the Product Rule, in conjunction with the Chain Rule, we compute the derivative as follows: and. So the slopes of the tangent lines to the curve

1. By the Product Rule, in conjunction with the Chain Rule, we compute the derivative as follows: and. So the slopes of the tangent lines to the curve MAT 11 Solutions TH Eam 3 1. By the Product Rule, in conjunction with the Chain Rule, we compute the derivative as follows: Therefore, d 5 5 d d 5 5 d 1 5 1 3 51 5 5 and 5 5 5 ( ) 3 d 1 3 5 ( ) So the

More information

Review: A Cross Section of the Midterm. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Review: A Cross Section of the Midterm. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Review: A Cross Section of the Midterm Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the limit, if it eists. 4 + ) lim - - ) A) - B) -

More information

e x for x 0. Find the coordinates of the point of inflexion and justify that it is a point of inflexion. (Total 7 marks)

e x for x 0. Find the coordinates of the point of inflexion and justify that it is a point of inflexion. (Total 7 marks) Chapter 0 Application of differential calculus 014 GDC required 1. Consider the curve with equation f () = e for 0. Find the coordinates of the point of infleion and justify that it is a point of infleion.

More information

Tangent Line Approximations. y f c f c x c. y f c f c x c. Find the tangent line approximation of. f x 1 sin x

Tangent Line Approximations. y f c f c x c. y f c f c x c. Find the tangent line approximation of. f x 1 sin x SECTION 9 Differentials 5 Section 9 EXPLORATION Tangent Line Approimation Use a graphing utilit to graph f In the same viewing window, graph the tangent line to the graph of f at the point, Zoom in twice

More information

1 The Derivative and Differrentiability

1 The Derivative and Differrentiability 1 The Derivative and Differrentiability 1.1 Derivatives and rate of change Exercise 1 Find the equation of the tangent line to f (x) = x 2 at the point (1, 1). Exercise 2 Suppose that a ball is dropped

More information

Additional Topics in Differential Equations

Additional Topics in Differential Equations 6 Additional Topics in Differential Equations 6. Eact First-Order Equations 6. Second-Order Homogeneous Linear Equations 6.3 Second-Order Nonhomogeneous Linear Equations 6.4 Series Solutions of Differential

More information

Vector-Valued Functions

Vector-Valued Functions Vector-Valued Functions 1 Parametric curves 8 ' 1 6 1 4 8 1 6 4 1 ' 4 6 8 Figure 1: Which curve is a graph of a function? 1 4 6 8 1 8 1 6 4 1 ' 4 6 8 Figure : A graph of a function: = f() 8 ' 1 6 4 1 1

More information

AP Calculus BC Summer Assignment 2018

AP Calculus BC Summer Assignment 2018 AP Calculus BC Summer Assignment 018 Name: When you come back to school, I will epect you to have attempted every problem. These skills are all different tools that we will pull out of our toolbo at different

More information

Mathematics Extension 2

Mathematics Extension 2 Student Number ABBOTSLEIGH AUGUST 007 YEAR ASSESSMENT 4 HIGHER SCHOOL CERTIFICATE TRIAL EXAMINATION Mathematics Etension General Instructions Reading time 5 minutes. Working time 3 hours. Write using blue

More information

PREPARATION FOR CALCULUS

PREPARATION FOR CALCULUS PREPARATION FOR CALCULUS WORKSHEETS Second Edition DIVISION OF MATHEMATICS ALFRED UNIVERSITY Contents Real Numbers Worksheet Functions and Graphs Worksheet 5 Polynomials Worksheet. Trigonometry Worksheet

More information

FIRST- AND SECOND-ORDER IVPS The problem given in (1) is also called an nth-order initial-value problem. For example, Solve: Solve:

FIRST- AND SECOND-ORDER IVPS The problem given in (1) is also called an nth-order initial-value problem. For example, Solve: Solve: .2 INITIAL-VALUE PROBLEMS 3.2 INITIAL-VALUE PROBLEMS REVIEW MATERIAL Normal form of a DE Solution of a DE Famil of solutions INTRODUCTION We are often interested in problems in which we seek a solution

More information

Name: Date: Period: Calculus Honors: 4-2 The Product Rule

Name: Date: Period: Calculus Honors: 4-2 The Product Rule Name: Date: Period: Calculus Honors: 4- The Product Rule Warm Up: 1. Factor and simplify. 9 10 0 5 5 10 5 5. Find ' f if f How did you go about finding the derivative? Let s Eplore how to differentiate

More information

VECTOR FUNCTIONS. which a space curve proceeds at any point.

VECTOR FUNCTIONS. which a space curve proceeds at any point. 3 VECTOR FUNCTIONS Tangent vectors show the direction in which a space curve proceeds at an point. The functions that we have been using so far have been real-valued functions. We now stud functions whose

More information

Law of Sines, Law of Cosines, Heron s Formula:

Law of Sines, Law of Cosines, Heron s Formula: PreAP Math Analsis nd Semester Review Law of Sines, Law of Cosines, Heron s Formula:. Determine how man solutions the triangle has and eplain our reasoning. (FIND YOUR FLOW CHART) a. A = 4, a = 4 ards,

More information

Chapter 27 AB Calculus Practice Test

Chapter 27 AB Calculus Practice Test Chapter 7 AB Calculus Practice Test The Eam AP Calculus AB Eam SECTION I: Multiple-Choice Questions DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. At a Glance Total Time 1 hour and 45 minutes Number

More information

9.1 Practice A. Name Date sin θ = and cot θ = to sketch and label the triangle. Then evaluate. the other four trigonometric functions of θ.

9.1 Practice A. Name Date sin θ = and cot θ = to sketch and label the triangle. Then evaluate. the other four trigonometric functions of θ. .1 Practice A In Eercises 1 and, evaluate the si trigonometric functions of the angle. 1.. 8 1. Let be an acute angle of a right triangle. Use the two trigonometric functions 10 sin = and cot = to sketch

More information

10.4 Nonlinear Inequalities and Systems of Inequalities. OBJECTIVES 1 Graph a Nonlinear Inequality. 2 Graph a System of Nonlinear Inequalities.

10.4 Nonlinear Inequalities and Systems of Inequalities. OBJECTIVES 1 Graph a Nonlinear Inequality. 2 Graph a System of Nonlinear Inequalities. Section 0. Nonlinear Inequalities and Sstems of Inequalities 6 CONCEPT EXTENSIONS For the eercises below, see the Concept Check in this section.. Without graphing, how can ou tell that the graph of + =

More information

APPM 1360 Final Exam Spring 2016

APPM 1360 Final Exam Spring 2016 APPM 36 Final Eam Spring 6. 8 points) State whether each of the following quantities converge or diverge. Eplain your reasoning. a) The sequence a, a, a 3,... where a n ln8n) lnn + ) n!) b) ln d c) arctan

More information

Math Exam 02 Review

Math Exam 02 Review Math 10350 Exam 02 Review 1. A differentiable function g(t) is such that g(2) = 2, g (2) = 1, g (2) = 1/2. (a) If p(t) = g(t)e t2 find p (2) and p (2). (Ans: p (2) = 7e 4 ; p (2) = 28.5e 4 ) (b) If f(t)

More information

11 PARAMETRIC EQUATIONS, POLAR COORDINATES, AND CONIC SECTIONS

11 PARAMETRIC EQUATIONS, POLAR COORDINATES, AND CONIC SECTIONS PARAMETRIC EQUATIONS, POLAR COORDINATES, AND CONIC SECTIONS. Parametric Equations Preliminar Questions. Describe the shape of the curve = cos t, = sin t. For all t, + = cos t + sin t = 9cos t + sin t =

More information

F dr y 2. F r t r t dt. a sin t a sin t a cos t a cos t a 2 cos 2 t a 2 sin 2 t. P dx Q dy yy. x C. follows that F is a conservative vector field.

F dr y 2. F r t r t dt. a sin t a sin t a cos t a cos t a 2 cos 2 t a 2 sin 2 t. P dx Q dy yy. x C. follows that F is a conservative vector field. 6 CHAPTER 6 VECTOR CALCULU We now easil compute this last integral using the parametriation given b rt a cos t i a sin t j, t. Thus C F dr C F dr Frt rt dt a sin ta sin t a cos ta cos t a cos t a sin t

More information

First-Order Linear Differential Equations. Find the general solution of y y e x. e e x. This implies that the general solution is

First-Order Linear Differential Equations. Find the general solution of y y e x. e e x. This implies that the general solution is 43 CHAPTER 6 Differential Equations Section 6.4 First-Order Linear Differential Equations Solve a first-order linear differential equation. Solve a Bernoulli differential equation. Use linear differential

More information

1.5. Analyzing Graphs of Functions. The Graph of a Function. What you should learn. Why you should learn it. 54 Chapter 1 Functions and Their Graphs

1.5. Analyzing Graphs of Functions. The Graph of a Function. What you should learn. Why you should learn it. 54 Chapter 1 Functions and Their Graphs 0_005.qd /7/05 8: AM Page 5 5 Chapter Functions and Their Graphs.5 Analzing Graphs of Functions What ou should learn Use the Vertical Line Test for functions. Find the zeros of functions. Determine intervals

More information

Derivatives 2: The Derivative at a Point

Derivatives 2: The Derivative at a Point Derivatives 2: The Derivative at a Point 69 Derivatives 2: The Derivative at a Point Model 1: Review of Velocit In the previous activit we eplored position functions (distance versus time) and learned

More information

Part 1: Integration problems from exams

Part 1: Integration problems from exams . Find each of the following. ( (a) 4t 4 t + t + (a ) (b ) Part : Integration problems from 4-5 eams ) ( sec tan sin + + e e ). (a) Let f() = e. On the graph of f pictured below, draw the approimating

More information

Introducing Instantaneous Rate of Change

Introducing Instantaneous Rate of Change Introducing Instantaneous Rate of Change The diagram shows a door with an automatic closer. At time t = 0 seconds someone pushes the door. It swings open, slows down, stops, starts closing, then closes

More information

UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives

UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives Chapter 3 3Quadratics Objectives To recognise and sketch the graphs of quadratic polnomials. To find the ke features of the graph of a quadratic polnomial: ais intercepts, turning point and ais of smmetr.

More information

All work must be shown in this course for full credit. Unsupported answers may receive NO credit.

All work must be shown in this course for full credit. Unsupported answers may receive NO credit. AP Calculus. Worksheet All work must be shown in this course for full credit. Unsupported answers ma receive NO credit.. What is the definition of a derivative?. What is the alternative definition of a

More information

Chapter 1 Prerequisites for Calculus

Chapter 1 Prerequisites for Calculus Section. Chapter Prerequisites for Calculus Section. Lines (pp. ) Quick Review.. + ( ) + () +. ( +). m. m ( ) ( ). (a) ( )? 6 (b) () ( )? 6. (a) 7? ( ) + 7 + Yes (b) ( ) + 9 No Yes No Section. Eercises.

More information

Directional derivatives and gradient vectors (Sect. 14.5). Directional derivative of functions of two variables.

Directional derivatives and gradient vectors (Sect. 14.5). Directional derivative of functions of two variables. Directional derivatives and gradient vectors (Sect. 14.5). Directional derivative of functions of two variables. Partial derivatives and directional derivatives. Directional derivative of functions of

More information

11.1 Inverses of Simple Quadratic and Cubic Functions

11.1 Inverses of Simple Quadratic and Cubic Functions Locker LESSON 11.1 Inverses of Simple Quadratic and Cubic Functions Teas Math Standards The student is epected to: A..B Graph and write the inverse of a function using notation such as f (). Also A..A,

More information

Quick Review 4.1 (For help, go to Sections 1.2, 2.1, 3.5, and 3.6.)

Quick Review 4.1 (For help, go to Sections 1.2, 2.1, 3.5, and 3.6.) Section 4. Etreme Values of Functions 93 EXPLORATION Finding Etreme Values Let f,.. Determine graphicall the etreme values of f and where the occur. Find f at these values of.. Graph f and f or NDER f,,

More information

CHAPTER 3: DERIVATIVES

CHAPTER 3: DERIVATIVES (Exercises for Section 3.1: Derivatives, Tangent Lines, and Rates of Change) E.3.1 CHAPTER 3: DERIVATIVES SECTION 3.1: DERIVATIVES, TANGENT LINES, and RATES OF CHANGE In these Exercises, use a version

More information

PreCalculus Second Semester Review Chapters P-3(1st Semester)

PreCalculus Second Semester Review Chapters P-3(1st Semester) PreCalculus Second Semester Review Chapters P-(1st Semester) Solve. Check for extraneous roots. All but #15 from 1 st semester will be non-calculator. P 1. x x + 5 = 1.8. x x + x 0 (express the answer

More information