5.5 Worksheet - Linearization

Transcription

1 AP Calculus 4.5 Worksheet 5.5 Worksheet - Linearization All work must be shown in this course for full credit. Unsupported answers ma receive NO credit. 1. Consider the function = sin. a) Find the equation of the tangent line when = 0. b) Graph both equations on our calculator in a standard viewing window. When would the tangent line be a good approimation for the curve? (Tr zooming in at the origin) c) Use the tangent line to approimate sin (0.). [Wh is using the tangent line from part a a good approimation?]. The approimate value of 4 sin at = 0.1, obtained from the tangent to the graph at = 0, is [Wh does the question ask ou to use the tangent line at = 0?] A.00 B.03 C.06 D.1 E.4 3. Use linearization to approimate f (5.0) if f Find the error for our approimation.

2 4. Suppose ou were asked to determine the value of a) Use linearization to approimate the value of b) Use differentials to approimate the value of [With calculator] Let f be the function given b f 3. The tangent line to the graph of f at = is used to approimate the values of f (). Which of the following is the greatest value for which the error resulting from this tangent line approimation is less than 0.5? A.4 B.5 C.6 D.7 E.8 6. Find the differential d when d = 0. and = 1, if e. Eplain what ou ve found. 1, find d if 3 and d. Eplain what ou ve found. 7. If sin 3 10

3 8. Without a calculator, use differentials to approimate v0 9. [Calculator Required] The range R of a projectile is R sin 3 and is the angle of elevation. Let 0 00, where v feet per second and let change from 10 to 11. [Remember in calculus, all degree measurements should be in radians] a) Find the actual change in the range. v 0 is the initial velocit in feet per second b) Use differentials to approimate the change in the range. What is the error in our approimation? 10. The radius of a ball bearing is measured to be 0.7 inch. If the measurement is correct to within 0.01 inch, estimate the error in the volume of the ball bearing.

4 AP Calculus 4.1 Worksheet 5.1 Worksheet - Etreme Values of Functions All work must be shown in this course for full credit. Unsupported answers ma receive NO credit. 1. Earlier this ear we had the Intermediate Value Theorem (IVT) and now we have the Etreme Value Theorem (EVT). The hpothesis of each theorem is what is needed to appl each theorem. a) What is the hpothesis of the IVT (i.e. what is needed to appl the IVT)? b) What is the hpothesis of the EVT?. Using the graphs provided, find the minimum and maimum values on the given interval. If there is no maimum or minimum value, eplain which part of the hpothesis of the Etreme Value Theorem is not satisfied. (a) [ 1, ] (b) [ 1, ] (c) ( 1, ) Ma: Ma: Ma: Min: Min: Min: 3. When looking for etrema, where do ou find the candidates for the candidates test? Each of the following statements is NOT ALWAYS TRUE. Eplain/Show wh each statement is false. 4. If f '( 5) = 0, then there is a maimum or a minimum at = If = is a critical number, then f ' ( ) = An etrema occurs at ever critical number. 7. If m is a local minimum and M is a local maimum of a continuous function, then m < M.

5 8. If f is a continuous, decreasing function on [0, 10] with a critical point at (4, ), which of the following statements MUST BE FALSE? A f (10) is an absolute minimum of f on [0, 10] B f (4) is neither a relative maimum nor a relative minimum f '4 does not eist. C ( ) D f '4 ( ) = 0 E f '4 ( ) < 0 9. Find the etrema on each interval and where the occur Use a candidates test. 1 a) f ( ) = + ln when b) g( ) ln ( 1) = + when 0 < < 3 c) k( ) = 5 when 3 < < Find the etrema of h( q) = sinq- cos( q) for 0 q p. Use our graphing calculator to investigate first. 11. [No Calculator] An open-top bo is to be made b cutting congruent squares of side length from the corners of a 5- b 8-inch sheet of tin and bending up the sides (see figure below). a) Write an equation for the Volume of the bo. b) What is the domain of this function? 5" c) How large should the squares be to maimize the volume? 8" d) What are the dimensions of the bo with maimum volume? What is the maimum volume?

6 AP Calculus 4. Worksheet 5. Worksheet - Increasing/Decreasing Functions & Critical Points All work must be shown in this course for full credit. Unsupported answers ma receive NO credit. 1. State the hpothesis of each of the following theorems: a) IVT b) EVT c) MVT. State the MVT two different was a) in words b) algebraicall 3. For each of the following, (a) state whether or not the function satisfies the hpotheses of the MVT on the given interval, and (b) if it does, find the value of c that the MVT guarantees. a) f 14 1 on the interval [1, 6] b) h 1 3 on [ 1, 1] 4. When a trucker came to his second toll booth in a 169-mile stretch of road, he handed in a ticket stub that was stamped hours earlier. The trucker was cited for speeding. Wh? 5. Suppose f () is a differentiable function on the interval [ 7, 1] such that f ( 7) = 4 and f (1) = 1. a) Eplain wh f must have at least one value in the interval ( 7, 1), where the function equals. 5 b) Eplain wh there must be at least one point in the interval ( 7, 1) whose derivative is Make a sign chart for the following functions:

7 a) f b) g Summarize how we will use calculus to determine whether a function is increasing or decreasing. 8. For each function, determine where the function is increasing or decreasing. Then, find the value of an relative etrema. Justif ALL answers. a) h b) 3 f 6 15 c) k 4

8 9. [Calculator] The Profit P in dollars made b a fast food restaurant selling hamburgers is given b P , a) Find the intervals on which P is increasing or decreasing. [Use Calculus!] b) Find the maimum profit. 10. If g is a differentiable function such that g () < 0 for all real numbers, and if f ' 9 g following is true? A) f has a relative maimum at = 3 and a relative minimum at = 3. B) f has a relative minimum at = 3 and a relative maimum at = 3. C) f has relative minima at = 3 and at = 3. D) f has relative maima at = 3 and at = 3. E) It cannot be determined if f has an relative etrema., which of the 11. A 16-m pea patch is to be enclosed b a fence and divided into two equal parts b another fence parallel to one of the sides. What dimensions for the outer rectangle will require the smallest total length of fence? How much fence will be needed?

9 1. A car is traveling on a straight road. For 0 t 4 seconds, the car s velocit v(t), in meters per second, is modeled b the piecewise-linear function defined b the graph below. a) For each of v '4 and v ' 0, find the value or eplain wh it does not eist. Indicate units of measure. b) Let a(t) be the car s acceleration at time t, in meters per second per second. For 0 < t < 4, write a piecewise-defined function for a(t). Velocit (meters per second) (0, 0) v(t) (4, 0) (16, 0) Time (sec) (4, 0) t c) Find the average rate of change of v over the interval 8 < t < 0. Does the Mean Value Theorem guarantee a value of c, for 8 < c < 0, such that v' c is equal to this average rate of change? Wh or wh not? 13. Use the graph of f ' defined on [0, 6] provided below to estimate the following: a) When is f increasing? When is f decreasing? Justif our response. b) Determine the -coordinates of all local etrema. Justif our response. 14. [No Calculator Allowed] Let f be a function defined on the closed interval 3 < < 4 with f (0) = 3. The graph of f ', the derivative of f, consists of one line segment and a semicircle, as shown below. a) On what intervals, if an, is f increasing? Decreasing? Justif our answer. b) Find all values of for which f assumes a relative maimum. Justif our answer. Graph of

10 AP Calculus 4.3 Worksheet 5.3 Worksheet - Connecting f' and f'' with the Graph of f All work must be shown in this course for full credit. Unsupported answers ma receive NO credit. 1. Complete each statement with the correct word. a) When f ' is, the graph of f is increasing,. b) When f ' is, the graph of f is decreasing,. c) When f '' is, the graph of f is concave upward. d) When f '' is, the graph of f is concave downward. e) When f ' is, the graph of f is concave upward. f) When f ' is, the graph of f is concave downward.. Use the function [No calculator allowed] a) Where is the function increasing? Justif our response. b) Where is the function decreasing? Justif our response. c) Where is the function concave up? Justif our response. d) Where is the function concave down? Justif our response. e) Where are the point(s) of inflection? Justif our response. f) Find ALL etrema and justif our response. g) Create a sketch of the function using the information ou have found from a f.

11 3. Find all local etrema of the function and justif our response using the nd derivative test Determine the intervals on which the graph of each function is concave up or concave down and determine all points of inflection. Justif our responses. a) b) If f is continuous on [0, 3] and satisfies the following: 0 0 < < < < < < 3 3 f f ' DNE 3 f '' 0 1 DNE 0 a) Find the absolute etrema of f and where the occur. Justif our response. b) Find an points of inflection. Justif our response. c) Sketch a possible graph of f.

12 6. Let f be a function that is continuous on the interval [0, 4]. The function f is twice differentiable ecept at =. The function f and its derivatives have the properties indicated in the table above, where DNE indicates that the derivatives of f do not eist at =. 0 0 < < < < < < < < 4 f 1 Negative 0 Positive Positive 0 Negative f ' 4 Positive 0 Positive DNE Negative 3 Negative f '' Negative 0 Positive DNE Negative 0 Positive a) Describe the behavior of f () in each interval using the information above. 0 < < 1 1 < < < < 3 3 < < 4 f () b) For 0 < < 4, find all values of at which f has a relative etremum. Determine whether f has a relative maimum or a relative minimum at each of these values. Justif our answer. c) On the aes provided, sketch the graph of a function that has all the characteristics of f. 7. [Calculator Required] Let f be a function defined for 0 with f (0) = 5 and f ', the first derivative of f, given b f ' e 4 sin. The graph of f ' is shown below. a) Use the graph of f ' to determine whether the graph of f is concave up, concave down, or neither on the interval 1.7 < < 1.9. Eplain our reasoning. b) Write an equation for the line tangent to the graph of f at the point (, 5.63). Graph of f '

13 8. [No Calculator Allowed] Let f be a function defined on the closed interval 3 < < 4 with f (0) = 3. The graph of f ', the derivative of f, consists of one line segment and a semicircle, as shown below. a) On what intervals, if an, is f increasing? Decreasing? Justif our answer. b) Find all values of for which f assumes a relative maimum. Justif our answer. c) Where is the graph of f concave up? concave down? Justif our answers. Graph of d) Find the -coordinate of each point of inflection of the graph of f on the open interval 3 < < 4. Justif our answer. e) Find an equation for the line tangent to the graph of f at the point (0, 3). f) Sketch a possible graph of f. 9. Suppose that at an time t (sec) the current I (amp) in an alternating current circuit is I = cos t + sin t. What is the peak (largest magnitude) current for this circuit? Justif our response.