Derivatives Worksheet 1 - Understanding the Derivative Math G180 Classwork. Solve the problem.

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1 Derivatives Worksheet 1 - Understanding the Derivative Math G180 Classwork Solve the problem. 1) Find f'() using the limit definition of the derivative given f() = ) Find g'(z) using the limit definition of the derivative given g(z) = z 2-5 z + 7,. 3) Find f'() using the limit definition of the derivative given f() = ) Find f'() using the limit definition of the derivative given f() = ) Use the limit definition of the derivative to find f'() given f() = ) A ball thrown verticall upward at time t = 0 (seconds) has height (t) = 96t - 16t 2 (ft) at time t. a) Graph (t) b) Find the velocit function, v(t), using the limit definition of the derivative. c) What is the velocit of the ball at t = 2 sec.? d) What is the velocit of the ball at t = 4 sec.? What does the negative sign mean? e) When is the velocit function equal to 0? What phsical events are occurring at each of the times when the velocit is 0? 7) A ball thrown verticall upward at time t = 0 (seconds) has height h(t) = 64t - 16t 2 (ft) at time t. The velocit function is v(t) = 64-32t a) Graph h(t) b) Use 2nd calc ma to find the time, tma, when the ball reaches its maimum height, and illustrate the point ( tma, hma) on our graph. c) Using the velocit function, v(t), evaluate v( tma). d) Do ou think the ball indeed comes to a stop instantaneousl before coming back down? Wh? Find the slope of the line tangent to the graph at the given point. Use the limit definition of the derivative, or numerical methods (small h =.001), to find the slope. 8) = 8 + 7, = -2 9) = , = -2 1

2 10) For the graph of f() below, estimate f'() at the given values b using the grids and the fact that f'() = slope of the tangent line at the point a) = -2 b) = -1 c) = -0.4 d) = 0 e) = 1 f) = 2 g) = 3.5 h) = 5 Estimate the slope of the curve at the indicated point. 11) 12) 13) 2

3 14) 15) 16) 17) The table gives the position of a particle moving along the -ais as a function of time in seconds, where is in angstroms. What is the average velocit of the particle from t = 2 to t = 8? 18) The figure below shows a particle s distance from a point. What is the particle s average velocit from t = 1 to t = 3? 3

4 19) For the function shown in the figure below, at what labeled points is the slope of the graph positive? Negative? At which labeled point does the graph have the greatest (i.e., most positive) slope? The least slope (i.e., negative and with the largest magnitude)? 20) The figure below shows the graph of f. Match the derivatives in the table with the points a,b,c,d,e. 21) Suppose that f() is a function with f(100) = 35 and f '(100) = 3. Estimate f(102). 22) The function in the figure below has f(4) = 25 and f '(4) = 1.5. Find the coordinates of the points A, B, C. 4

5 23) Consider the function shown in the figure below. (a) Write an epression involving f for the slope of the line joining A and B. (b) Draw the tangent line at C. Compare its slope to the slope of the line in part (a). (c) Are there an other points on the curve at which the slope of the tangent line is the same as the slope of the tangent line at C? If so, mark them on the graph. If not, wh not? 24) The population, P(t), of China, in billions, can be approimated b P(t) = 1.267(1.007) t where t is the number of ears since the start of According to this model, how fast was the population growing at the start of 2000 and at the start of 2007? Use t =.001 to do the approimations and give our answers in millions of people per ear. Solve the problem. 25) The graph of = f() in the accompaning figure is made of line segments joined end to end. Graph the derivative of f. (3, 5) (6, 5) (-3, 2) (-5, 0) (0, -1) 5

6 26) Use the following information to graph the function f over the closed interval [-5, 6]. i) The graph of f is made of closed line segments joined end to end. ii) The graph starts at the point (-5, 1). iii) The derivative of f is the step function in the figure shown here. So the step function shown is the derivative of f(), ou draw the graph of the original f()

7 The graph of a function is given. Choose the answer that represents the graph of its derivative. 27) A) 15 B) C) 15 D)

8 28) A) 15 B) C) 15 D) ) The figure below is the graph of f ', the derivative of a function f. On what interval(s) is the function f (not f ' ) (a) Increasing? (b) Decreasing? 8

9 30) A child inflates a balloon, admires it for a while and then lets the air out at a constant rate. If V (t) gives the volume of the balloon at time t, then the figure below shows V '(t) as a function of t. At what time does the child: (a) Begin to inflate the balloon? (b) Finish inflating the balloon? (c) Begin to let the air out? 31) A vehicle moving along a straight road has distance f(t) from its starting point at time t. Which of the graphs in the figure below could be f '(t) for the following scenarios? (Assume the scales on the vertical aes are all the same.) (a) A bus on a popular route, with no traffic (b) A car with no traffic and all green lights (c) A car in heav traffic conditions 32) Note: An Alternate Notation for the Derivative: d d is another notation for f '(), the derivative. It comes from which ou recognize as the slope from Algebra I. (You now realize that the "slope" in Algebra I was just the average rate of change of a function from one point to another!) As we take the limit as goes to 0, this difference quotient becomes the derivative at a point. So to remind us of where the derivative came from, we can use the d d notation. But there is also another reason for using this notation: It reminds us we are differentiating the function with respect to the variable (as opposed to another variable that ma be involved in the problem which we will see later). Another notation: If = f() then ' denotes the derivative of f(). And f ' denotes the derivative of f(). 33) The cost, C (in dollars), to produce g gallons of a chemical can be epressed as C = f(g). Using units, eplain the meaning of the following statements in terms of the chemical: (a) f(200) = 1300 (b) f '(200) = 6 9

10 34) The temperature, T, in degrees Fahrenheit, of a cold am placed in a hot oven is given b T = f(t), where t is the time in minutes since the am was put in the oven. (a) What is the sign of f '(t)? Wh? (b) What are the units of f '(20)? What is the practical meaning of the statement f '(20) = 2? 35) Investing $1000 at an annual interest rate of r%, compounded continuousl, for 10 ears gives ou a balance of $B, where B = g(r). Give a financial interpretation of the statements: (a) g(5) (b) g '(5) 165. What are the units of g '(5)? 36) In 2011, the Greenland Ice Sheet was melting at a rate between 82 and 224 cubic km per ear. (a) What derivative does this tell us about? Define the function and give units for each variable. (b) What numerical statement can ou make about the derivative? Give units. 37) On Ma 9, 2007, CBS Evening News had a 4.3 point rating. (Ratings measure the number of viewers.) News eecutives estimated that a 0.1 drop in the ratings for the CBS Evening News corresponds to a $5.5 million drop in revenue. Epress this information as a derivative. Specif the function, the variables, the units, and the point at which the derivative is evaluated. 38) Let W be the amount of water, in gallons, in a bathtub at time t, in minutes. (a) What are the meaning and units of dw/dt? (b) Suppose the bathtub is full of water at time t0, so that W(t0) > 0. Subsequentl, at time tp > t0, the plug is pulled. Is dw/dt positive, negative, or zero: (i) For t0 < t < tp? (ii) After the plug is pulled, but before the tub is empt? (iii) When all the water has drained from the tub? 39) In Ma 2007 in the US, there was one birth ever 8 seconds, one death ever 13 seconds, and one new international migrant ever 27 seconds. (a) Let f(t) be the population of the US, where t is time in seconds measured from the start of Ma Find f '(0). Give units. (b) To the nearest second, how long did it take for the US population to add one person in Ma 2007? 40) During the 1970s and 1980s, the build up of chlorofluorocarbons (CFCs) created a hole in the ozone laer over Antarctica. After the 1987 Montreal Protocol, an agreement to phase out CFC production, the ozone hole has shrunk. The ODGI (ozone depleting gas inde) shows the level of CFCs present. Let O(t) be the ODGI for Antarctica in ear t; then O(2000) = 95 and O '(2000) = Assuming that the ODGI decreases at a constant rate, estimate when the ozone hole will have recovered, which occurs when ODGI = 0. 10

11 41) Multiple choice: g(v) is the fuel efficienc, in miles per gallon, of a car going at a speed of v miles per hour. What is the practical meaning of g '(55) = 0.54? There ma be more than one option. (a) When the car is going 55 mph, the rate of change of the fuel efficienc decreases to approimatel 0.54 miles/gal. (b) When the car is going 55 mph, the rate of change of the fuel efficienc decreases b approimatel 0.54 miles/gal. (c) If the car speeds up from 55 mph to 56 mph, then the fuel efficienc is approimatel 0.54 miles per gallon. (d) If the car speeds up from 55 mph to 56 mph, then the car becomes less fuel efficient b approimatel 0.54 miles per gallon. 42) For the function graphed below, give the signs of the first and second derivative. 43) Graph the second derivative of the function shown below: 44) At eactl two of the labeled points in the figure below, the derivative f ' is 0 ; the second derivative f " is not zero at an of the labeled points. On a cop of the table, give the signs of f, f ', f " at each marked point. 11

12 45) Graph the functions described in parts (a) (d). (a) First and second derivatives everwhere positive. (b) Second derivative everwhere negative; first derivative everwhere positive. (c) Second derivative everwhere positive; first derivative everwhere negative. (d) First and second derivatives everwhere negative. 46) A particle is thrown upward from the ground. Sketch the graph of the height of the particle against time if velocit is positive and acceleration is negative. 47) At which of the marked -values in the figure below can the following statements be true? Assume 2 and 4 are NOT infection points. (a) f() < 0 (b) f ' () < 0 (c) f() is decreasing (d) f ' () is decreasing (e) Slope of f() is positive (f) Slope of f() is increasing 48) a) Eplain what is wrong with each statement: i. A function that is not concave up is concave down. ii. When the acceleration of a car is zero, the car is not moving. b) Give an eample of: i. A function that has a non-zero first derivative but zero second derivative. ii. A function for which f '(0) = 0 but f "(0) 0. 49) True or false and wh? a) If f "() > 0 then f '() is increasing. b) The instantaneous acceleration of a moving particle at time t is the limit as h -> 0 of (v(t+h)-v(t))/h c) A function which is monotonic on an interval is either increasing or decreasing on the interval. d) The function f() = 3 is monotonic on an interval. e) The function f() = 2 is monotonic on an interval. Solve the problem. 50) Sketch a continuous curve = f() with the following properties: f(2) = 3; f () > 0 for > 4; and f () < 0 for < 4. 12

13 51) The graph below shows the first derivative of a function = f(). Select a possible graph of f that passes through the point P. f P A) B) C) D) 13

14 52) The graph below shows the first derivative of a function = f(). Select a possible graph f that passes through the point P. f P A) B) C) D) 53) Decide if f() = ( + ) is differentiable at = 0. Tr zooming in on our graphing calculator, or calculating the derivative f '(0) from the limit definition, or both. 14

15 54) For the following functions, discuss an continuit or differentiabilit problems the ma have Given the graph of f, find an values of at which f 55) is not defined. 56) 57) 58) 15

16 59) 60) 61) Find the derivative using the short-cut rules. 62) = Find the derivative. 63) = ) = ) s = 3t 2 + 4t ) w = z -5-1 z 67) = Find the second derivative. Graph the derivative and graph the 2nd derivative. Illustrate that the 2nd derivative is giving ou the slope values of the derivative function. 68) = Find the second derivative. 69) =

17 70) s = 2t ) = Solve the problem. 72) The position of a bod moving on a coordinate line is given b s = t 2-6t + 7, with s in meters and t in seconds. When, if ever, during the interval 0 t 6 does the bod change direction? 73) At time t, the position of a bod moving along the s-ais is s = t 3-15t t m. Find the bod's acceleration each time the velocit is zero. 74) At time t 0, the velocit of a bod moving along the s-ais is v = t 2-9t + 8. When is the bod moving backward? 75) A ball dropped from the top of a building has a height of s = t 2 meters after t seconds. How long does it take the ball to reach the ground? What is the ball's velocit at the moment of impact? 76) A rock is thrown verticall upward from the surface of an airless planet. It reaches a height of s = 120t - 2t 2 meters in t seconds. How high does the rock go? How long does it take the rock to reach its highest point? 77) The area A =!r 2 of a circular oil spill changes with the radius. At what rate does the area change with respect to the radius when r = 9 ft? 78) The driver of a car traveling at 60 ft/sec suddenl applies the brakes. The position of the car is s = 60t - 3 t 2, t seconds after the driver applies the brakes. How far does the car go before coming to a stop? 79) Find the equation of the line tangent to the graph of f at (1,1), where f is given b f() = Then graph the function and the tangent line together - see that our tangent line is the perfect fit to the curve) 80) Using a graph to help ou, find the equations of all lines through the origin tangent to the parabola = Sketch the lines on the graph. 81) A ball is dropped from the top of the Empire State building to the ground below. The height,, of the ball above the ground (in feet) is given as a function of time, t, (in seconds) b = t 2. (a) Find the velocit of the ball at time t. What is the sign of the velocit? Wh is this to be epected? (b) Show that the acceleration of the ball is a constant. What are the value and sign of this constant? (c) When does the ball hit the ground, and how fast is it going at that time? Give our answer in feet per second and in miles per hour (1 ft/sec = 15/22 mph). 17

18 82) At a time t seconds after it is thrown up in the air, a tomato is at a height of f(t) = 4.9t 2 +25t+3 meters. (a) What is the average velocit of the tomato during the first 2 seconds? Give units. (b) Find (eactl) the instantaneous velocit of the tomato at t = 2. Give units. (c) What is the acceleration at t = 2? (d) How high does the tomato go? (e) How long is the tomato in the air? 83) If M is the mass of the earth and G is a constant, the acceleration due to gravit, g, at a distance r from the center of the earth is given b g = GM r 2 (a) Find dg/dr. (b) What is the practical interpretation (in terms of acceleration) of dg/dr? Wh would ou epect it to be negative? (c) You aretold that M = and G = where M is in kilograms and r in kilometers. What is the value of dg/dr at the surface of the earth (r = 6400 km)? (d) What does this tell ou about whether or not it is reasonable to assume g is constant near the surface of the earth? 84) The graph of = has a slope of 5 at two points. Find the coordinates of the points. 85) Find the equation of the line tangent to = at = 2. 86) An animal population is given b P(t) = 300 (1.044) t where t is the number of ears since the stud of the population began. Find P'(5) and interpret our result. 87) With a earl inflation rate of 5%, prices are given b P = P0(1.05) t where P0 is the price in dollars when t = 0 and t is time in ears. Suppose P0 = 1. How fast (in cents/ear) are prices rising when t = 10? 88) For what value(s) of a are = a and = 1+ tangent at = 0? Eplain. 89) True or False: a) If f() is increasing, then f '() is increasing. b) There is no function such that f '() = f() for all. c) If f() is defined for all, then f '() is defined for all. Find ' using the product rule. 90) = (3-3 )( ) 91) = (2-3 )(2 + 1) 18

19 92) z = 3 2 e Find the derivative of the function using the quotient rule. 93) = ) = ) #3-27 odd. 19

20 Composition practice to get read to find the derivative of a composite function. Find the requested composition or operation. 96) f() = , g() = 4-1 Find (f g)(). 97) f() = Find (g f)()., g() = Perform the requested composition or operation. 98) Find (g f)(4) when f() = 9-6 and g() =

21 Use the graphs to evaluate the epression. 99) (f g)(-2) = f() 5 = g() ) (f g)(0) = f() 5 = g() (2, 4) 3 3 (-2, 2) 2 (2, 2) 2 (1, 2) (-1, -1) -1 (1, -1) (0, -2) (-1, -2) (-2, -4) ) g(f(4)) = f() = g()

22 Use the tables to evaluate the epression if possible. 102) Find (g f)(10) f() g() ) Find the derivatives for 1-31 odd. 22

23 104) Derivatives of Trig functions. Consider the graph of f() = sin() below a) Make a table of estimated slope values of the graph of f() for values incrementing b 0.5 on the interval from = 0 to 6.5 b) Graph the derivative of f() using our data from (a). What classic trig function does it look like? 105) Another wa to estimate the graph of the derivative of f() = sin() (or an function) is to graph the difference quotient with h =.001, or other small h value, in our TI as follows: 1 = (f(+.001)-f())/.001. So in the case of f() = sin() that would be: 1= (sin(+.001)-sin())/.001 Use this method to graph the derivative of f() = sin() and report which trig function results. 106) Here is a limit that is critical to the proof that the derivative of sin() is cos(). Consider lim sin(h) h 0 h a) Find the limit b graphical methods, that is, graph 1= sin()/ (remember, our calculator must be in radian mode, not degrees) on : -1 to 1 ; : 0 to 2 and cop to our paper. Use 2nd calc value and put in =.001. What is our estimate for the limit using this "h" value of.001? b) Evaluate using smaller h values. What do ou conclude the limit is? 107) Here is another limit that is critical to the proof that the derivative of sin() is cos(). Consider lim 1-cos(h) h 0 h a) Find the limit b graphical methods, that is, graph 1= (1-cos())/ (remember, our calculator must be in radian mode, not degrees) on : -1 to 1 ; : -1 to 1 and cop to our paper. Use 2nd calc value and put in =.001. What is our estimate for the limit using this "h" value of.001? b) Evaluate using smaller h values. What do ou conclude the limit is? 108) Prove d d (tan()) = 1 cos 2 b re-writing tan() as sin() cos() and then using the quotient rule to find the derivative. 109) Estimate the graph of the derivative of f() = tan() using the difference quotient with h =.001 in our TI as follows: 1= (tan(+.001)-tan())/.001 Then put in 2 = 1/cos()^2. What do ou notice? 23

24 110) Find the derivatives for 3-23 odd. 111) A boat at anchor is bobbing up and down in the sea. The vertical distance,, in feet, between the sea floor and the boat is given as a function of time, t, in minutes, b = 15 + sin(2!t). (a) Find the vertical velocit, v, of the boat at time t. (b) Make rough sketches of and v against t. 24

25 112) An oscillating mass, m, at the end of a spring is at a distance from its equilibrium position given b = A sin k m t The constant k measures the stiffness of the spring and A is a constant. (a) Find a time at which the mass is farthest from its equilibrium position. Find a time at which the mass is moving fastest. Find a time at which the mass is accelerating fastest. (b) What is the period, T, of the oscillation? (c) Find dt/dm. What does the sign of dt/dm tell ou? Solve the problem. 113) A rocket is launched verticall and is tracked b a radar station located on the ground 6 mi from the launch pad. Suppose that the elevation angle θ of the line of sight to the rocket is increasing at 5 per second when θ = 60. What is the velocit of the rocket at this instant? 114) Write an equation of the line that is tangent to the curve = sin at the point P with -coordinate! ) Derivatives of ln(), a, Arctan(), Arcsin() a) d) 1-1 d d ( 2 ) = b) d d (Arctan()) = e) d d (ln()) = 1 c) d d (a ) = ln(a)a d d (Arcsin()) = ) Show that d d (Arcsin()) = 1 b using the formula for the derivative of the inverse of a function. Hint: 1-2 Arcsin() is the inverse of sin(), and the epression cos(arcsin() is found in terms of b using a right triangle picture as in the video for the proof of d d (Arctan()) = above. 117) Find d (Arccos()) b using the formula for the derivative of the inverse of a function (and the right triangle d picture method shown in the video above). 25

26 118) Find the derivatives in 1-23 odd. Use implicit differentiation to find d/d. 119) 2-2 = 1 120) = 8 121) + 1 = 4 At the given point, find the slope of the curve or the line that is tangent to the curve, as requested 122) 6 6 = 64, slope at (2, 1) 123) 5 2 -! cos = 6!, slope at (1,!) 124) = , tangent line at (0, 1) 26

27 125) Find d d using implicit differentiation for 1-19 odd. 126) Find d d given point using implicit differentiation for 23, 25 and find the slope of the tangent line to the curve at the Find the linearization L() (the tangent line) of f() at = a AND evaluate the difference between L and f at a point nearb. That is, find L(a + 0.1) - f(a + 0.1) to see how far off the linear approimation is from the actual function value if we move 0.1 awa to the right. Graph f() and L() to verif if our difference result makes sense. 127) f() = , a = ) f() = , a = 0 129) f() = sin, a = 0 27

28 130) Do 1-7 odd. In each case graph the function and it's tangent line (local linearization) on the same graph. 131) This problem shows how local linearization (the tangent line approimation) is used in the real world: Air pressure at sea level is 30 inches of mercur. At an altitude of h feet above sea level, the air pressure, P, in inches of mercur, is given b P =30e -3.23X10-5 h (a) Sketch a graph of P against h. (b) Find the equation of the tangent line at h = 0. (c) A rule of thumb used b travelers is that air pressure drops about 1 inch for ever 1000-foot increase in height above sea level. Write a formula for the air pressure given b this rule of thumb. (d) What is the relation between our answers to parts (b) and (c)? Eplain wh the rule of thumb works. (e) Are the predictions made b the rule of thumb too large or too small? Wh? Find the value or values of c that satisf the equation for the function and interval. 132) f() = , [-3, 2] f(b) - f(a) b - a = f (c) in the conclusion of the Mean Value Theorem 133) f() = ln ( - 3), [4, 6] Round to the nearest thousandth. Solve the problem. 134) Show that the function f() = satisfies the hpothesis of the mean value theorem on the interval [-1, 2]. Find all numbers c in that interval that satisf the conclusion of that theorem. 135) Show that the function f() = 2-5 satisfies the hpothesis of Rolle's theorem on the interval [0, 5], and find all numbers in (0, 5) that satisf the conclusion of that theorem. Determine whether the function satisfies the hpothesis of the Mean Value Theorem for the given interval. 136) f() = 1/3, -4,2 137) g() = 3/4, 0,5 28

29 138) A trucker handed in a ticket at a toll booth showing that in 2 hours he had covered 230 miles on a toll road with speed limit 65 mph. The trucker was cited for speeding. Wh? Using the derivative of f() given below, determine the intervals on which f() is increasing or decreasing. 139) f () = 1/3 ( - 6) 140) f () = ( - 1) e - 141) f () = ( + 5) 2 e - 29

30 Answer Ke Testname: DERIVATIVES WORKSHEET 1 - NATALIE (1) 1) ) a) 2z - 5 3) 6-4 4) ) f'() = 3 6) b) v(t) = 96-32t, c) 32 ft/sec., d) -32 ft/sec, the ball is going down. 7) b) tma = 2 sec., and hma = 64 ft., c) v(2) = 0 ft/sec, d) Yes, it does, because... ou answer wh in class! 8) m = 8 9) m = 2 10) The following were obtained b plugging into the actual f '() function, so ours will be approimate: a) 5.6 b) 1.8 c).096 d) -.8 e) -2.2 f) -2.4 g) -.45 h) ) 0 12) 1 13) -1 14) -1 15) Undefined 16) 2 17) 3 angstroms/sec 18) 2 meters/sec 19) Positive: A and D ; Negative: C and F ; Most positive: A; Most negative: F 20) f '(d) = 0, f '(b) = 0.5, f '(c) = 2, f '(a) = 0.5, f '(e) = 2 21) About 41 Here's wh: f '(100) = 3 means that at = 100 f() increases 3 units per 1 unit change in. (3 = 3/1). But = 102 is 2 units awa from 100 so f() will change approimatel two increments of 3, so 2(3) = 6 units. So = 41. Note: The instantaneous rate of change, 3, is onl eact right at = 100, but assuming f() does not fluuate wildl around that point then it is safe to sa at = 102 that f() will increase 6 units to be about ) (4,25); (4.2,25.3); (3.9,24.85) 23) (a) (f(b) f(a))/(b a) (b) Slopes same (c) Yes, about two thirds of the wa from c and b. 24) 8.84 million people/ear; 9.28 million people/ear (our answers ma be slightl different) 25)

31 Answer Ke Testname: DERIVATIVES WORKSHEET 1 - NATALIE (1) 26) (-3, 5) (3, 5) (-5, 1) (0, 2) (6, -1) 27) B 28) B 29) see video 30) (a) t = 3 (b) t = 9 (c) t = 14 31) a) graph II b) graph I c) graph III 32) 33) (a) Costs $1300 for 200 gallons (b) It costs about $6 for the 201 st gallon, that is, when the are producing 200 gallons the rate of change is $6 per gallon, so if the produce 1 more gallon at that 200 gallon level of production, that would cost about $6. It is "about" because the instantaneous rate of change is $6 but increasing 1 gallon is a g = 1 and so that is 1 unit awa from 200 which is close but not eactl at ) (a) Positive, because the temperature of the am is alwas increasing (well, until it reached the oven temperature then it would hold there, so f '(t) = 0 from that time and after). (b) (degrees F)/min ; At the 20 minute mark the temperature of the am is increasing at a rate of 2 degrees F per minute. (So at the 21 minute mark the am would be about 2 degrees F hotter than it was at the 20 minute mark) 35) a) If the interest rate is 5% then the balance will be about $1649 at the end of the 10 ears. b) When the interest rate is 5% the balance would increase b about $165 per 1 percent increase in the interest rate. That is, an etra percentage point would ield an increase of about $165. That is, if the interest rate was upped 1% to 6% then at the end of the 10 ears the balance would be about $165 more than it would have been if the interest rate was 5%. ; The units are dollars/%. 36) a) f(t) = amount of cubic km of ice at t ears after 2000 (the 2000 is arbitrar - ou could pick an ear). The derivative of f(t), f '(t), is the rate at which the ice is melting t ears after the ear b) f '(11) is between -82 cubic km per ear and -224 cubic km per ear. The units of the derivative are (cubic km)/ear. So in 2012 the ice sheet would have lost between 82 and 224 cubic km of ice. 37) f(r) = revenue in millions for a viewer rating point r. f '(r) = rate of change of the revenue (in million dollars) per rating point. The units of f '(r) are (million dollars)/(rating point). For the given scenario we have: f '(4.3) = (-5.5 million dollars)/(-.1 points) = $55 million per point. So we learn from this that if the CBS news show had a 4.3 point rating then if the rating would increase to 4.4 then CBS would gain $5.5 million and if the rating decreased to 4.2 then CBS would lose $5.5 million. 38) (a) Gal/minute (b) (i) 0 (ii) Negative (iii) 0 39) (a) people/sec (b) 12 seconds 40) Find the linear equation for O(t): (2000,95) is a point and the slope is So O(t) - 95 = -1.25(t ). So we get O(t) = -1.25t Set this equal to 0, get t = In the ear 2076 the hole will be closed (if the rate of decrease of the inde stas at per ear) 31

32 Answer Ke Testname: DERIVATIVES WORKSHEET 1 - NATALIE (1) 41) b and d 42) see video 43) see video 44) Discuss in class 45) Discuss in class 46) Discuss in class. 47) a) 4, 5 b) 3, 4 c) 3, 4 d) 2, 3 e) 1, 2, 5 f) 1, 4, 5 48) Discuss in class 49) Discuss in class 50) Answers will var. A general shape is indicated below: 51) A 52) C 53) see video 54) 1) discontinuous at = 1, not differentiable at = 2 and 3. 2) continuous everwhere, not differentiable at = 2 and 4. 3) continuous everwhere, not differentiable at the sharp corner on the left and the one on the right. 4) continuous and differentiable everwhere. 55) = 0 56) = -2, 2 57) = 0 58) = 0 59) Defined for all values of 60) = 0 61) = 0, 3 62) ) ) ) 6t ) -5z z 2 32

33 Answer Ke Testname: DERIVATIVES WORKSHEET 1 - NATALIE (1) 67) ) 12 69) ) 4t 71) ) t = 3 sec 73) a(6) = 6 m/sec 2, a(4) = -6 m/sec 2 74) 1 < t < 8 75) 4 sec, -128 m/sec 76) 1800 m, 30 sec 77) 18" ft 2 /ft 78) 300 ft 79) 80) 81) (a) v(t) = 32t, v 0 because the height is decreasing (b) a(t) = 32 (c) t = 8.84 seconds, v = mph 82) a) 15.2 m/sec b) 5.4 m/sec c) -9.8 m/sec 2 (that's (m/sec)/sec = the change in meters per second ever second, and, -9.8 m/sec 2 is the acceleration due to gravit. ) d) ma height occurs when velocit = 0. That occurs at t = sec. so ma height = meters (rounded off) e) sec. (2nd calc zero on the f(t) function ou have in 1 gets this. 83) (a) dg/dr = 2GM/r 3 (b) dg/dr is rate of change of acceleration. g decreases with distance (c) (d) Magnitude of dg/dr small; reasonable 84) At (-1,7) and (7,-209) 85) = ) At the 5 ear mark the animal population was increasing at a rate of about 16 animals per ear. 87) 7.95 cents/ear 88) at e. 89) a) False, eample: f() = ln() b) False f() = e c) False f() = 90) ) ) dz d = 6e e 93) = /2 94) = ( - 1) 2 33

34 Answer Ke Testname: DERIVATIVES WORKSHEET 1 - NATALIE (1) 95) answers: 96) ) 98) ) 0 100) ) 4 102) 21 34

35 Answer Ke Testname: DERIVATIVES WORKSHEET 1 - NATALIE (1) 103) Answers: 104) a) show table b) = cos(). 105) = cos() 106) a) b) 1 107) a).0005 b) 0 108) Just do it - it works out! 109) = 1 cos 2 is the resulting graph, the estimated graph of the derivative of tan() and the graph of 1 cos 2 coincide. 35

36 Answer Ke Testname: DERIVATIVES WORKSHEET 1 - NATALIE (1) 110) answers 111) answers 112) answers: 113) 2! 3 mi/s 7540 mi/h 114) = 115) see the links 116) discuss in class 36

37 Answer Ke Testname: DERIVATIVES WORKSHEET 1 - NATALIE (1) ) ) answers: 119) - 120) ) - 2( + 1) 122) ) -2! 124) = ) answers: 126) 23) slope is infinite 25) -23/9 go to the links to obtain the graphs. 37

38 Answer Ke Testname: DERIVATIVES WORKSHEET 1 - NATALIE (1) 127) L() = , L(a + 0.1) - f(a + 0.1) = ) L() = 2 + 6, L(a + 0.1) - f(a + 0.1) = ) L() =, L(a + 0.1) - f(a + 0.1) = ) answers: 131) answers: 132) ) ) c = ±1 135) f(0) = 0 = f(5); f'() = 2-5; c = ) No 137) Yes 138) As the trucker's average speed was 115 mph, the Mean Value Theorem implies that the trucker must have been going that speed at least once during the trip. 139) Decreasing on (0, 6); increasing on (-, 0) (6, ) 140) Decreasing on (-, 1); increasing on (1, ) 141) Never decreasing; increasing on (-, -5) (-5, ) 38