PROPERTIES OF EXPONENTIAL FUNCTIONS AND LOGS

Transcription

1 PROPERTIES OF EXPONENTIAL FUNCTIONS AND LOGS

2 Calculus Lesson- Derivatives of Eponentials and Logs Name: Date: Objective: to learn how to find the derivative of an eponential or logarithmic function Finding the Derivative of f ( ) e Using limit process: RULE: Eamples: Find f () for each: 6. f ( ) 3e. f ( ) e 3 3. e f ( ) f ( ) e 5. f ( ) ( e 4) 6. f ( ) ( e ) f ( ) e e f ( ) e f ( )

3 Calculus WKST: Differentiating e Name: Date: Find dy. Do not leave comple fractions in your answers. d. y e. y sec e y 3 4. y = cos e e 3. e 3y 5 e 5. y = e d 6. sin e d d e 7. d 8. y 4 Solve for dy/d (hint: convert into e ) 9. f ( ) e 4 Find the equation of the line tangent to the function when = 0. y e e ( ) Solve for dy/d. ) Find the derivative of y of the function y e 3 3

4 Finding the Derivative of f ( ) ln Eamples:. f() = 4 ln. f ( ) ln 3. f ( ) (8 3ln ) 3 4. f ( ) ln(5 ) 5. f ( ) 3 ln(4 ) 6. f ( ) (ln ) 4 7. For what values of does f ( ) ln( ) have a horizontal tangent line? Putting it all together Find f () for f ( ) 6 9. Find f () for f ( ) log 8(3 ) 4

5 Calculus WKST: Differentiating and Integrating Logs Name: Date: Find the derivative of these epressions ) ln ) ln 5 3) 4ln 4) ln 5) 3 ln 6) ln 7) b ln 8) ln( 4 ) 9) ln(3 5 ) 3 0) ln(sin ) ) ln (sin ) ) ln 3) sec()d 4) 8tan(4 )d 5) (csc() 5)d 6) cot(5t)dt 7) 4 d 8) d 3 9 F() is the number of errors that occur in a book when = the number of pages in the book. F() 0 (t 8t )dt a) How many errors will be in the book if it had 8 pages? b) What is the average number of errors per page if the book has 9 pages? c) What is the maimum and minimum number of errors the book can have if it can have a maimum of 45 pages in it? 5

6 Application Problems:. The percent of information retained t months after being tested on that information is given by: f(t) = 8 4 ln (t + ). Evaluate f() and f () and interpret your answers.. The cost and revenue functions for producing units of a certain product are C() = and R() = 8 ln(. + ). a. Find the marginal cost if 00 units are produced. b. Find the marginal revenue if 00 units are produced. c. Find the marginal profit if 00 units are produced. d. Graph the two functions and use your graphing calculator to find the break-even points. e. To make a profit, how many units should be produced. 6

7 Calculus Lesson- Antiderivatives and Indefinite Integrals Name: Date: Anti-derivative = Anti-differentiation = Working backwards Eample: is an antiderivative of The general form of the antiderivative is given as F() + C, where C is any constant, called the constant of integration. Any antiderivatives of f will differ only by a constant. Notation: Let y = dy + C. Then This is called a differential equation, an equation that involves d, y, and a derivative of y. We can re-write this equation as dy d. This is called the differential form of the equation. In general, Let y F( ) and F( ) f ( ). dy Then f() is a differential equation and d dy f() d We write this using integral notation: is the differential form of the equation. y f ( ) d F( ) C is the indefinite integral sign; it means the antiderivative of f with respect to. f() is the integrand, the function to be antidifferentiated. = the variable of integration C = the constant of integration Rules of Integration 0 d k d kf () d n d cos d sin d sec d csc d sec tan d = csc cot d 7

8 Steps for Integrating: () Re-write the original integral into a form in the table. () Integrate (take the antiderivative). (3) Simplify. Eamples. d. 4 d Find the general solution for differential 5cos d dr equation. d d 6. sec d 7. d 8. t dt 8

9 Calculus Lesson- Antiderivatives and Initial Conditions and Particular Solutions Name: Date: Objective- To learn how to find anti-derivatives involving particular values and initial conditions. dy The general solution to the differential equation 3 is d 3 dy (3 ) d y d If we are given the value of y for one value of this is called an initial condition then we can get a particular solution; that is, we can find a particular constant C and write a specific antiderivative as our answer. Eamples. Given F( ) 3 d and F() 4. Find the particular solution. F() =. Find g() if g( ) 6 and g(0) = A ball is thrown upward with an initial velocity of 64 ft/sec from an initial height of 80 ft. Using the fact that the acceleration due to gravity is 3 ft/sec², (a) find the position function giving the height s as a function of time t. (b) When does the ball hit the ground? 9

10 4. The rate of growth dp/dt of a population of bacteria is proportional to the square root of t, where P is the dp population size and t is the time in days 0 t 0. That is, k t. The initial size of the dt population is 500. After day the population has grown to 600. Estimate the population after 7 days. 5. The lift in a tall building passes the 50th floor with a velocity of -8m/s and an acceleration of t 5m / s. If each floor spans a height of 6m find at which floor the lift will stop. 9 0

11 Calculus Lesson- Definite Integrals Name: Date: Old Stuff: Important Theorem: The Definite Integral as the Area of a Region If f is continuous and nonnegative on the closed interval [a, b], then the area of the region bounded by the graph of f, the -ais, and the vertical lines = a and = b is given by: Area b a f ( ) d. Important Properties of Definite Integrals () If f is defined at = a, then a a f ( ) d 0. () If f is integrable on [a,b], then f ( ) d f ( ) d. (Reversing the limits of integration reverses the sign.) a b (3) Additive Interval Property: If f is integrable on the closed intervals determined by a, b, and c, with b c b a c b,then: f ( ) d f ( ) d f ( ) d. a a c b a (4) Constant Multiple Property: If f is integrable on [a, b] and k is a constant, then: b a kf ( ) d k f ( ) d. b a (5) Additive Function Property: If f and g are integrable on [a, b], then b b b f ( ) g( ) d f ( ) d g( ) d. a a a (6) Preservation of Sign Property: If f is integrable and nonnegative on [a, b], then b 0 f ( ) d. a (7) Preservation of Inequality Property: If f and g are integrable on [a, b], and f ( ) g( ) for every in [a, b], then b a f ( ) d g( ) d. b a

12 Eamples of Use: Eample : Given d 60 and d, evaluate 3 d 4 5d d Repeat Eample using the Fundamental Theorem of Calculus. 3 d 4 5d d Repeat Eample using a GC approach 3 d 4 5d d

13 3 6 Eample : Given f ( ) d 4 and f ( ) d, evaluate a. f ( ) d b f ( ) d 3 c. f ( ) d d f ( ) d 3 Properties of Definite Integrals Practice Eample 3: Given the shaded region A has an area of.5, and (a) f ( ) d (b) f ( ) d f ( ) d 3.5, evaluate the following integrals. (c) 6 0 f ( ) d (d) f ( ) d 0 (e) The average value of f over the interval [0,6] is 3

14 Eample 4: Being given that 0 sind and using what you know about integrals, graphing curves, and using a little bit of intuition, determine the values of the following integrals: a. b. c. sin d 0 sin d / 0 sin d d. ( sin ) d e. sind f. 0 0 sin( ) d g. sin udu h. sin d i. 0 cos d 0 Eample 5: Use areas to evaluate the integral. 4 d Eample 6: Use areas and your knowledge of discontinuous functions to find d 4

15 Calculus More Practice with Definite Integrals Name: Date:. Suppose Find each of the following integrals, if possible: (a) 4 4 f ( ) d 5, f ( ) d, h( ) d 7 f ( ) d (b) f ( ) d 4 (c) [ f ( ) 3h( )] d (d) f ( ) d 0 (e) h( ) d (f) f ( ) d h( ) d 4. Suppose (a) 5 f ( ) d 4, f ( ) d 6, g( ) d 8 g( ) d (b) g( ) d 5 5 (c) 3 f ( ) d (d) f ( ) d 5 (e) 5 [ f ( ) g( )] d (f) [ 4 f ( ) g( )] d 5 5

16 Calculus Lesson- Net versus Total Area Name: Date: Objective: To learn the difference between net and total area. Find the net and total area of each function over the given interval analytically:. y = 3 4, [-,]. y = sin, [0,3π/] Find the total area of each shaded region

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18 Integration as Net Change E. A car moving with initial velocity of 5 mph accelerates at the rate of a(t) =.4t mph/sec for 8 sec. (a) How fast is the car going when the 8 seconds are up? (Instantaneous Rate) (b) How far did the car travel during those 8 seconds? (Net Change) Practice:. v(t) = 5 cos t 0 < t < π (a) determine when the particle is moving to the left, to the right, and stopped (b) find the particle s displacement for the given time interval (c) find the total distance traveled by the particle 8

19 . The following is a graph of velocity. The particle starts at = when t = 0. (a) Find where the particle is at the end of the trip (b) Find the total distance traveled by the particle Suppose that a tank initially contains 000 gal of water and that the rate of change of its volume after the tank drains for 5 min is V (t) = 0.8t 40 (in gallons per min). How much water does the tank contain after it has been draining for a half-hour? 4. Suppose that the population in Juneau in 970 was 5 (in thousands) and that its rate of growth t years later was P (t) = t t (in thousands per year). What was its population in 990? 9

20 Calculus WKST- More Practice with Antiderivatives Name: Date: Part I: Evaluate the integrals below: a) for f() = 0 3, determine F() e 5 b) for f() =, determine F(). 4 5 e c) for f() =, determine F(). 3 d) for f() = -3 ( - ), determine F(). 0

21 Part II: The function v(t) = 0.(t 5) gives the speed of a rocket in kilometers per second for the interval 5 < t < 0. i) How quickly is the rocket moving after 0 seconds? ii) How quickly is the rocket accelerating after 0 seconds? iii) How far has the rocket traveled after 0 seconds? Estimate your answer using delta = one second and both the left and right rectangle methods. iv) Estimate this same distance using the trapezoid method. v) How far does the rocket travel between 0 and fifteen seconds? Estimate using the same criteria. vi) How does your answer compare to the algebraic calculation of the definite integral?

22 Calculus Lesson- Integration by Substitution Name: Date: This is a technique that allows us to integrate a great many functions by putting them in the form of some function we have in our table of integrals. By the Chain Rule, d F ( g ( )) F ( g ( )) g ( ), so d by the Fundamental Theorem, F( g( )) g( ) d F( g( )) C. If we consider u = g(), some function of, then du g() d and f(u)du F(u) C. Steps for Integrating by Substitution- Indefinite Integrals. Choose a substitution u = g(), such as the inner part of a composite function.. Compute du g() d. 3. Re-write the integral in terms of u and du. 4. Find the resulting integral in terms of u. 5. Substitute g() back in for u, yielding a function in terms of only. 6. Check by differentiating. Eamples. d. 3 d

23 3. cos sin d d 5. Solve the differential equation: dy 4 d 8 Integration by Substitution Indefinite Integrals with Initial Conditions Use the method above to find a general solution, then use the initial condition to find C and a particular solution. 6. Find a function f that satisfies f ( ) sec ( ) and whose graph passes through the point,. 7. Find f if f ( ) 8 and f() = 7. 3

24 Integration by Substitution Definite Integrals Change of Variables for Definite Integrals: If the function u = g() has a continuous derivative on [a, b] and f is continuous on the range of g, then b a gb ( ) f ( g( )) d f ( u) du g( a) Steps for Integrating by Substitution Definite Integrals. Choose a substitution u = g(), such as the inner part of a composite function.. Compute du g() d. Compute new u-limits of integration g(a) and g(b). 3. Re-write the integral in terms of u and du, with the u-limits of integration. 4. Find the resulting integral in terms of u. 5. Evaluate using the u-limits. No need to switch back to s! d d d e. e e e d 4

25 Calculus Lesson- Slope Fields, Separation of Variables & Growth/Decay Name: Date: Objective: To learn how to generate slope fields. To learn how to solve homogeneous differential equations using separation of variables. Eample a: Eample b: Suppose that you know that the point (0, -) is on a particular solution of the differential equation above. By following slopes, draw on the diagram above what you think the particular solution looks like. Note: The graph should follow the pattern of the slope field but may go between the points rather than through them. Eample c: Solve the differential equation dy y from the previous eample by first separating the variables. Find the d particular solution that contains the point given in the last eample. Does your solution make sense when compared to the graph of the slope field? 5

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28 Recall: Directly Proportional: Inversely Proportional: Jointly: f () k f () k f (,y) ky 3. Let P(t) represent the number of wolves in a population at time t years when t 0. The population P(t) is increasing at a rate directly proportional to 800 P(t), where the constant of proportionality is k. a) At P(0) 500, find P(t) in terms of t and k. b) Use your answer to part (a) to determine k if P()=700. c) P(t) lim t 4. Solve the differential equation: dp dt k 0 t 5. Solve the differential equation that models the verbal statement: a. The rate of change of P with respect to t is proportional to 5 t. b. The rate of change of y with respect to varies jointly as and L y. 8

29 6. Solve the differential equation: dy d 4 7. Solve the differential equation: dy d 4 y 8. The number of bacteria in a culture is increasing continuously following the model y Ce kt. There are 5 bacteria in the culture after hours and 350 bacteria in the culture after 4 hours. a. Find the initial population. b. Write an eponential growth model for the bacteria population. Let t represent time in hours. c. Use the model to determine the number of bacteria after 8 hours. d. After how many hours will the bacteria count be 5,000? 9. Newton s Law of Cooling: the rate of change of the temperature in an object is proportional to the difference between the object s temperature and the temperature of the surrounding medium. A -ounce glass bottle of liquid is placed in a freezer that is kept at a constant temperature of 0F. The initial temperature of the liquid is 70F. After five minutes, the liquid s temperature is 50F. How much longer will it take the liquid to decrease to 37F? 9

30 Calculus Practice- Word Problems Name: Date: Newton s Law of Cooling: The rate of change in the temperature of an object is proportional to the difference between the temperature of the object and the temperature of the surrounding medium. E. Suppose a room is kept at a constant temperature of 60 and an object cooled from 00 to 90 in ten minutes. How much longer will it take for its temperature to decrease to 80? Work the following on notebook paper. Use your calculator and give decimal answers correct to three decimal places.. A pie is removed from an oven at 450 and left to cool at a room temperature of 70. After 30 minutes, the pie s temperature is 00. How many minutes after being removed from the oven will the temperature of the pie be 00?. A certain population increases at a rate proportional to the square root of the population. If the population goes from 500 to 3600 in five years, what is the population at the end of t years? 3. Water leaks out of a barrel at a rate proportional to the square root of the depth of the water at that time. If the water level starts at 36 in. and drops to 35 in. in one hour, how long will it take for all of the water to leak out of the barrel? 4. A student studying a foreign language has 50 verbs to memorize. The rate at which the student can memorize these verbs is proportional to the number of verbs remaining to be memorized, 50 y, where the student has memorized y verbs. Assume that initially no verbs have been memorized and suppose that 0 verbs are memorized in the first 30 minutes. (a) How many verbs will the student memorize in two hours? (b) After how many hours will the student have only one verb left to memorize? 5. At time t, t 0, the volume of a sphere is increasing at a rate proportional to the reciprocal of the radius. At t = 0, the radius of the sphere is, and at t = 5, the radius is. (a) Find the radius of the sphere as a function of t. (b) At what time t will the volume of the sphere be 7 times the volume at t = 0? Answers to Worksheet on Applications of Diff. Equations. 7.0 min.. 5. (a) 4 r t (b) t = 80 y (t 50) hrs 4. (a) 43.5 verbs (b) 3.89 hrs 30

31 Calculus Lesson- Homogeneous Differential Equations Name: Date: Objective: To verify Homogeneity and to solve Homogeneous Differential Equations. A homogeneous differential equation is an equation of the form: M(,y)d + N(,y)dy = 0 where M and N are homogeneous functions of the same degree. How to determine if functions are homogeneous (and determine the degree): Verify f (t,ty) t n f (,y) and state n. What this means: Take the original function and replace the & y with t & ty. Simplify and factor out a GCF. What remains should be t n the epression that is the original implicitly defined function. The n is the degree. How to solve a homogeneous differential equation: If M(,y)d + N(,y)dy = 0 is Homogeneous, then it can be transformed into a differential equation whose variables are separable by the substitution: y = v where v is a differentiable function of. Sub out an aspect of y using y = v Note: you will need to find y Simplify Separate variables and integrate Sub out any aspect of v using y=v Eamples:. Find the general solution of ( y )d 3ydy 0 3

32 . Solve the homogeneous differential equation: y' 3 y 3 y 3. Solve the homogeneous differential equation: y' 3y 3

33 Calculus Lesson- Area Between Curves Name: Date: How to find the area between two curves on [a, b]: Find the point(s) of intersection. Establish which function is dominant from a to the point of intersection. Integrate the difference between the dominant and non-dominant functions, respectively. Establish which function is dominant from the point of intersection to b. Integrate the difference between the dominant and non-dominant functions, respectively. Add these two positive areas together to get the total area. If looking for average: Divide your answer by b a.. Find the area of the region bounded between y 3 and y

34 . Find the area of the region bounded by y ln, the and y aes, and y 3 a. With respect to. b. With respect to y. 3. Find the area of the region in quadrant I bounded by y 8, y, and y 0 a. With respect to. b. With respect to y. 34

35 Calculus Lesson- Volumes of Cross Sections Name: Date: Objective: To find the volume of a solid given area boundaries of the base and known cross-sections DO NOW: Find the area of the region bounded by: y e, the aes, and the line = What if emerging from the area you calculated were infinitely many squares? What would be the volume of this new solid? Now try these. The base of solid S is the region enclosed by the graph of y ln, the line = e, and the -ais. If the cross-sections of S perpendicular to the -ais are semi-circles, then the volume of S is? 35

36 . The base of solid S is the region enclosed by the graphs f ( ), g( ), and = 0. If the cross-sections perpendicular to the -ais are equilateral triangles, find the volume of the solid. 3. Let R be the region enclosed by graphs of y ln and y = cos. a. Find the area of R (use your GC) b. The base of a solid is region R. Each cross section of the solid perpendicular to the -ais is an equilateral triangle. Write an epression involving one or more integrals that gives the volume of the solid. Do not evaluate. 36

37 Calculus Lesson- Volumes of Discs Name: Date: Objective: To find the volume of a solid using the disc method. DO NOW: Write an epression to find the area of the region bounded by the aes, y 3, and = 8 in Q What if this area was rotated around the -ais? What would be the volume of this new figure? this area was rotated around the line = 8? What would be the volume of this new figure? 37

38 Now try these. The region in the first quadrant bounded by the graphs of y = sec,, and the aes is rotated 4 about the -ais. What is the volume of the solid generated?. The volume of a solid obtained by revolving the region enclosed by the ellipse 9y 9 about the - ais is? 3. Let R be the region in the first quadrant bounded by the graphs of y, the line y = 6, and the y- ais. a. Find the area of R. b. Write, but do not evaluate, an integral epression that gives the volume of the solid generated when R is rotated about the line y = 6. 38

39 Calculus Lesson- Volumes of Washers Name: Date: Objective: To find the volume of a solid using the washer method. Let s get right to it What if you have a gap between the given area and the ais of revolution??? Let R be the region between the graphs of y = and y = sin from = 0 to =. What is the volume of the solid obtained by revolving R about the -ais? Now try these. The region enclosed by the graph of y, the line =, and the -ais is revolved about the y-ais. What is the volume of the solid generated? 39

40 . A region in the first quadrant is enclosed by the graphs of y e, =, and the coordinate aes. If the region is rotated about the y-ais, what is the volume of the solid generated? / 3 3. Let R be the region in the first quadrant is enclosed by the graphs of y, = 7, and the coordinate aes. If the region is rotated about the y-ais, what is the volume of the solid generated? 40

41 Calculus Lesson- Volumes of Cylindrical Shells Name: Date: Objective: To find the volume of a solid using the shell method. Let s get right to it Sometimes using the washer method can be a pain. This happens when you have to develop or more integrals to describe the bounded volume; instead of doing that, we can use the Cylindrical Shell Method. To do this we must create a rectangle PARALLEL to the ais of revolution and then find the volume of the CYLINDER that has that rectangle as its HEIGHT. The distance that the rectangle is away from the ais of revolution is the cylinder s RADIUS. So knowing those pieces of info and the fact that the volume of a cylinder is r h, we can now find the volume of a cylindrical shell. Eamples. If p is defined as the distance from the ais of revolution and the center of the rectangular strip then what is the radius of the outer cylinder?. What is the volume of the outer cylinder? 3. What is the radius and volume of the inner cylinder? 4. What is the volume of the Cylindrical Shell? 4

42 The Shell Method: Vertical Ais of Revolution Horizontal Ais of Revolution where p() or p(y) represent the radius of the cylindrical shell and h() or h(y) represent the height of the strip. 4

43 So let s revisit some eamples in the Washer Lesson and, this time, apply the Cylindrical Shell Method.. The region enclosed by the graph of y, the line =, and the -ais is revolved about the y-ais. What is the volume of the solid generated?. A region in the first quadrant is enclosed by the graphs of y e, =, and the coordinate aes. If the region is rotated about the y-ais, what is the volume of the solid generated? / 3 3. Let R be the region in the first quadrant is enclosed by the graphs of y, = 7, and the coordinate aes. If the region is rotated about the y-ais, what is the volume of the solid generated? 43

44 Calculus Practice- Volumes of Revolution Name: Date: Volume Problems- Definitely need to go to separate paper here! The goal is to find the VOLUME of a shape by adding up the volumes of slices. In general, the formula is b AREA ( ) d. a Eamples (Calculator Intensive). Find the volume of a solid created by revolving the line y = 4 on [, 4] over the -ais.. Find the volume of a solid created by revolving the lines y = e - and y = on [, 4] over the -ais Find the volume of the solid whose base is the region in quadrant I that is bounded by y, y 0, and. All cross-sections perpendicular to the -ais are a. Rectangles with height twice the width. b. Semi-circles. 4. The volume of the solid generated by rotating the region bounded by y 3, y 0, between 0and 4around the -ais 5. The volume of the solid generated by rotating the region in quadrant I bounded by y sin and y cos around the -ais. 6. The volume of the solid generated by rotating the region bounded by y 4, 0, and y 3 around the y-ais. 7. The volume of the solid generated by rotating the region in quadrant I bounded by, y, = and y 0 around a. The line y 0. b. The line 3. c. The line y. 44

45 Calculus Lesson- Applications of Integration: Arc Length, Surface Area Name: Date: Objective: To learn how to determine arc length of any function on [a, b] To learn how to find the area of the surface of a solid of revolution Definition of Arc Length: Let y = f() represent a smooth curve on the interval [a, b]. The arc length between a and b is: b s [ f '()] d a Also for a curve given by = g(y) on [c, d]: d s [g'(y)] dy c Eamples. Find the length of the arc connecting,y &,y on f() = m + b. Find the length of the arc of the graph of y 3 on the interval [0.5, ] 6 3. Find the length of the arc of the graph of y ln(cos ) on the interval 0, 4 45

46 Definition of the Area of a Surface of Revolution: Let y = f() have a continuous derivative on [a, b]. The area S of the surface of revolution formed by revolving the graph of f around a horizontal or vertical ais is: b S r() [ f '()] d a Also for a curve given by = g(y) on [c, d]: d S r(y) [g'(y)] dy c where r() is the distance between the graph of f and the ais of revolution. where r() is the distance between the graph of f and the ais of revolution. Eamples:. Find the surface area formed by revolving the graph of f () 3 on the interval [0,] about the -ais.. Find the surface area formed by revolving the graph of f () on [0, ] about the y-ais. 46

47 Calculus Lesson- Applications of Integration: Fluid Pressure and Fluid Force Name: Date: Objective: To learn how to use integration to determine fluid pressure and fluid force. Definition of Fluid Pressure The pressure on an object at depth h in a liquid is: Pressure = P = wh Where w is the weight-density of the liquid per unit of volume Here are some common weight-densities of fluids in pounds per square foot. Ethyl Alcohol 49.4 Gasoline Glycerin 78.6 Kerosene 5. Mercury Seawater 64.0 Water 6.4 Pascal s Principle states that the pressure eerted by a fluid at depth h is transmitted equally in all directions. Because fluid pressure is given in force per unit area (P = F/A), the fluid force on an object at a given depth would be the pressure times the area of the object: F PA There are two variations of this idea ) Horizontal orientation (downward force equivalent throughout) ) Vertical orientation (downward force varies depending on location) Good video eplanation: Eamples:. Find the fluid force on a rectangular metal sheet measuring 3 feet by 4 feet that is submerged horizontally in 6 feet of water. 47

48 The previous case was easy because it involved no calculus (no variation in downward force). The net case does involve calculus because the downward force varies depending on where on the vertically submerged surface you look. Definition of Force Eerted by Fluid (Vertical Case): F PA wha Where w is a constant We will use horizontal rectangles to determine the force at any depth. These rectangular areas are defined by the length of the rectangle (L) times the width ( d something ). Since the rectangles must be horizontal, the length would be L(y) and the width would be dy. The height (or depth) is known, therefore, as h(y). So we have the rule: F w d c h(y)l(y)dy where c represents the lowest vertical distance of the object and d represents the highest.. Find the fluid force on a rectangular metal sheet measuring 5 feet deep by foot wide that is submerged vertically such that the topmost point is foot below the surface of the water. 3. A vertical gate in a dam has the shape of an isosceles trapezoid 8 feet across the top and 6 feet across the bottom, with a height of 5 feet. What is the fluid force on the gate if the top of the gate is 4 feet below the surface of the water? 4. A triangular plate, base 5 feet, height 6 feet, is submerged in water, verte down, plane vertical, and feet below the surface. Find the total force on one face of the plate. 48

49 Calculus Lesson- Inverse Trig Functions, Differentiation, and Integration Name: Date: Objective: To learn how to find derivatives and anti-derivative of inverse trig functions You learned to use the following notation in Precalculus: cos arccos, sin arcsin, etc. You also learned that the inverse trig functions are restricted so that they will be functions. In Calculus the restrictions are understood to apply without our using a capital letter for the inverse. 0 arccos 0 arc sec (but ) 0 arc cot arcsin arc csc (but 0) arctan E. arcsin E. cos E. Use your calculator to find: arctan 0.3 (a) (b) csc.4 E. Evaluate: 4 (a) sin arccos 5 (b) sin arccos (c) sec arctan3 49

50 We can use implicit differentiation to derive the differentiation formulas for the inverse trig functions. d d arcsin u arctan u d d d d arcsec u Add to your Formula Sheet: d arcsin u d d d d d arctan u arcsec u d d arccos u d d arccot u d d arc cscu E. f arcsin f E. f arctan 3 f E. arc sec f e f E. f cos arcsin 3 f 50

51 E. Find an equation of the tangent line to the graph of where =. y arccos at the point on the graph We can also go the other way and find the integral: E. Evaluate: d 5 Problem Set Find the derivative.. f 3arcsin 5 5. arcsin y h arctan 3. y arccos g arcsec 3 7. p cos arcsin 4. arctan 5 f 8. q sec arctan 9. Given f arccos Find the line tangent to the function when = Find each integral: 3 d 0. d d 8 5

52 Calculus Lesson- Hyperbolic Trig Functions Name: Date: Objective: To learn the nature of Hyperbolic trig functions Definitions: e sinh e cosh e e e e tanh e e Using these relationships, find the following: Find which hyperbolic trig functions are even and odd and show their relationship. ) sinh( ) ) cosh( ) 3) tanh( ) Look at angle sum relationships. 4) sinh( ) 5) cosh( ) 6) tanh( ) Look at double angle relationships (hint: Use double angle identities) 7) sinh( ) 8) cosh( ) 9) tanh( ) 5

53 0) Look at the relationship between sinh ( ) and cosh ( ), tanh ( ) and sech ( ) and csch ( ) and coth ( ) ) Can we epress e and e - in terms of sinh and cosh? ) Find sinh - = 3) Find arcosh = 4) Find half angle results. sinh, cosh 53

54 Calculus Lesson- Hyperbolic Trig Function Graphs, Differentiation and Integration Name: Date: Objective: To learn how to graph Hyperbolic Trig Functions To learn how to find derivatives and anti-derivatives of Hyperbolic trig functions Graphing Hyperbolic Trig functions and Differentiation Graph each hyperbolic trig function then sketch the following transformations. ) y = sinh ( ) ) y = cosh ( 3) 3) y= tanh () + Differentiate each hyperbolic trig function using the eponential rules and simplify. 4) sinh() 5) cosh() 6) tanh() 54

55 7) sech() 8) coth() 9) csch() 0) d (cosh ) d d (sinh cosh( ) ) d ) Finally differentiate the following functions ) sinh - () 3) cosh - () 4) tanh - () 55

56 Integration of Hyperbolic trig functions. Find the integral of each hyperbolic trig function. ) sinh( ) d ) cosh( ) d Integrate : 3) tanh( ) d (hint, change into sinh and cosh) 4) sec h( ) d 56

57 If time 5) The equation of a catenary is f ( ) a cosh A catenary is the shape a rope would form if it were a fied at two points and allowed to hang free underneath those points. The St. Louis Arch is a catenary. The following equations will create the graph of the St. Louis Arch but upside down: f ( ) 75[cosh(0.009) ] and g( ) 68.8[cosh(0.0) ] a) Graph the system of equations that make up the arch on your calculator in a [-300,300] by [0,65] viewing window. b) Calculate the arc length of each arch for [-300,300], average them together and find the percent error given the actual arc length is 480ft. 57

58 Calculus Lesson- Integration using Algebraic Substitutions Name: Date: Objective: To learn about Integration Techniques involving: 3 Similar Integrals d d d Separating the numerator d A Substitution Using a u 6 d 6 58

59 Disguised Log Rule e d Disguised Power Rule tan ln(cos ) d Trig Identity Problems tan d 59

60 Calculus Lesson- Integration by Parts Name: Date: Objective: To learn how to integrate by parts Theorem: If u and v are functions of f and have continuous derivatives, then udv uv vdu Tips:. Try letting dv be the most complicated part of the integrand that fits a basic integration rule. Then u will be the remaining factor(s) of the integrand.. Try letting u be the portion of the integral whose derivative is a function simpler than u. Then dv will be the remaining factor(s) of the integrand. E. Find e d E. ln d E 3. Find arcsind E 4. 0 sin d 60

61 Problem Set. Evaluate.. e d. 4 e sin d 3. sec d 4. 0 e 5 d 5. e cos d 6. ln e e d 6

62 ln d arcsin 3 d d 3 0. e d e ln. arctan d d 6

63 Calculus Lesson- Integrals using Trig Functions Name: Date: Objective: To learn how to apply Trig Functions when finding integrals m n First we will look at integrals of the form: sin cos d. We will use the following identities: sin cos sin cos so cos sin cos cos sin so sin cos cos cos so cos Some helpful guidelines: ) If the power of cosine is odd and positive, save one cosine factor and convert the remaining factors to sine. ) If the power of sine is odd and positive, save one sine factor and convert the remaining factors to cosine. 3 cos 5 d E. E. cos 3 sin d 63

64 If the powers of both sine and cosine are even and nonnegative, use the identities: cos cos cos and sin E. sin 3cos 3 d 4 E. cos d You will also need to remember the Sum and Product Identities from Precalculus: sin Acos B sin A B sin A B sin Asin B cos A B cos A B cos Acos B cos A B cos A B E. sin 5cos 4 d 64

65 Now we ll look at integrals of the form m n m n sec tan d and csc cot d. You will need to remember: tan sec cot csc Ask yourself: Can I save a sec u or a secutan u? E. 5 tan sec d E. tan 4 3sec 3 d With cosecants and cotangents, ask yourself: Can I save a E. csc 4 4 cot d csc u or a cscucot u? 65

66 Calculus Lesson- Trig Substitutions Name: Date: Objective: To learn how to apply a trig substitution in finding integrals Trig substitution (or - substitution) can be used to evaluate integrals involving the radicals a u, a u, and u a. Our objective is to eliminate the radical in the integrand. ) For integrals involving a u, use sin. u variable a u radical Let sin cos a constant a constant u asin a u a cos ) For integrals involving a u, use tan. u variable a u radical Let tan sec a constant a constant u a tan a u asec 3) For integrals involving u a, use sec. u variable u a radical Let sec tan a constant a constant u asec u a a tan Notice that all start with: variable radical onefunction u other trig function a constant a constant 66

67 E. d 9 E. d 4 E. d 3 E. 3 3 d 67

68 Calculus Lesson- Integrals using Partial Fractions Name: Date: Objective: To learn how to integrate using partial fractions 4 4 E. d 30 E d E d 4 68

69 Problem Set ) d ) d 3 3) d 4) 3 d 5) 5 6) 4 d 3 d 69

70 Calculus Lesson- L Hopital s Rule and Improper Integrals Name: Date: Objective: To learn how to apply L Hopital s Rule To learn how to find improper integrals. I obtained the following from Paul s Online Math Notes. Everything is correct ecept the spelling So, L Hopital s Rule tells us that if we have an indeterminate form 0/0 or / all we need to do is differentiate the numerator and differentiate the denominator (separately) and reapply the limit. Improper Integrals L Hopital s Rule In problems -3, find the limit by: (a) using techniques from Chapter (b) using L Hopital s Rule.. lim 3 3. lim lim 0 sin 70

71 Evaluate by using L Hopital s Rule, if possible lim 3. lim ln 3 5. lim lim ln 0 6. lim 0 e 5. lim sin 7. lim 0 sin sin 3 6. lim 8. lim 0 arcsin 7. lim 0 9. lim lim 4 0. lim 9. 3 lim ln. lim 0. lim e 5. lim ln. lim 0 e e Answers to Worksheet on L Hopital s Rule e 4. DNE. (L Hop. doesn t work) (Not a L Hop. problem) 7

72 Improper Integrals These types of integrals are those that deal with an upper or lower limit that approaches infinity. In order to evaluate these limits we will look at an integral as the area under a curve and then calculate the equation that gives us the area under the curve. So looking at eample : d The way we approach this integral is the following. lim a a d Now the integral given is equivalent to a a Now if we take the limit lim a we find our answer. a We still need to deal with convergent and divergent limits however so be aware. Eample: d Show that this integral isn t a finite value. The three types of improper integrals are the following: f ( ) d f ( ) d f ( ) d a a When we split these integrals up, each integral must converge for the original integral to converge as well. 0 4 d e d 7

73 Problem Set- L Hopital and Improper Integrals Evaluate. cos. lim 0 5. lim 0 cos e. lim 0 lim 6. e 0 3. lim ln 7. lim cos cos 4. sin lim cos Evaluate d 3. 0 d 9. d d 0. ln e d 0 5. d 5 6. d e d. 4 d Answers to Worksheet on L Hopital s Rule and Improper Integrals converges to 5. converges to ln converges to 6. converges to diverges to converges to converges to e 3. converges to 3ln converges to 73