Differentiation Questions and Answers

2x 16x Simplify your answers Type an integer or a fraction

Problem 5 a Suppose that f x x 3 for x 0 and that f 1 1 Find f 4

You are a lifeguard and spot a distressed child 70 meters along the shore and 40 meters from the shore to the child You run along the shore and for a while and then jump into the water and swim from there directly to child You can run at a rate of 4 meters per second and swim at a rate of 1 2 meters per second How far along the shore should you run before jumping into the water in order to get to the child in the least amount of time Round your answer to three decimal places Hint Consider this diagram where a represents the distance running and y represents the distance swimming Try to write y in terms of x and use that to build the time as a function of x child you X 70 y 40

Read the following lines and answer the question that follows I can t wait to hear from you Which of the following is used to convey the same in the formal style OI look forward hearing from you O I am eager to hear from you O I think you will reply soon O I cannot wait to hear from you

Which of the following is are NOT TRUE O Narrative writing tells a made up story O Narrative nonfiction is also called biographical writing First person pronouns are used in biographies O Personal narratives talk about real events and people

The graph of f not f is given below At which of the marked values of a is A f x greatest x B f x least x c f x greatest x D f x least x E f x greatest F f x least x x1 x2 x3 x4 x5 x6 X

The hyperbolic cosine function cosh x is defined as follows cosh x Find the sixth degree Taylor Polynomial To x for cosh 4x about x 0 To x x 2 x 1 x6 e ex 2 X x5

Find the degree 3 Taylor polynomial Ts a of the function about 2 T3 x f x 3x 2 2

Rewrite the following equation in logarithmic form using a base 10 logarithm Use log for the base 10 logarithm b 10

A particular computing company finds that its weekly profit in dollars from the production and sale of x laptop computers is P x 0 007x 0 2x 700x 900 Currently the company builds and sells 11 laptops weekly a What is the current weekly profit b How much profit would be lost if production and sales dropped to 10 laptops weekly c What is the marginal profit when x 11 d Use the answer from part a and c to estimate the profit resulting from the production and sale of 12 laptops weekly

Using a calculator or computer generate a graph like the one shown below by graphing y 2 16 8x 8 and y 2x for 1 x 1 What is the relationship between y 2 16 8x 8 and y 2x Use your graph to estimate to one decimal place the largest magnitude of the error in approximating 2 16 8x 8 by 2x for 1 x 1 error Is the approximation an over or an underestimate when x 0 Enter over or under BEB

The graph of f not f is given below At which of the marked values of is A f x greatest x B f x least x c f x greatest D f x least x E f x greatest F f x least x x1 x2 x3 x4 x5 x6 X

The figure below shows a function f x At exactly two of the labeled points f x 0 At all of the labeled points f x 0 Select the correct signs for each of f f and f at each marked point Point f f f A A B A C E D E A

3x 3 Follow the steps for graphing a rational function to graph the function R x 5x 25 OC The function has three vertical asymptotes The leftmost asymptote is rightmost asymptote is Type equations Use integers or fractions for any numbers in the equations OD The function has no vertical asymptote the middle asymptote is and the Determine the behavior of the graph on either side of any vertical asymptotes if one exists Select the correct choice below and if necessary fill in the answer box es to complete your choice OA It approaches on one side of the asymptote s at x and o on the other It approaches either co or on both sides of the asymptote s at x Type an integer or a simplified fraction Use a comma to separate answers as needed Type each answer only once OB It approaches either or o on both sides of the asymptote s at x Type an integer or a simplified fraction Use a comma to separate answers as needed Type each answer only once OC It approaches o on one side of the asymptote s at x and o on the other Type an integer or a simplified fraction Use a comma to separate answers as needed Type each answer only once 4 OD There is no vertical asymptote

37 3i 5 1 39 3 21 4

The area of a healing wound is given by A The radius is decreasing at the rate of 3 millimeter per day at the moment when r 49 How fast is the area decreasing at that moment square millimeters per day

y 2e 2x 3x 6 y 0 5 olve the initial value problem above Do not include y in your answer

5 24 Find the general indefinite integral 5 3x 4x 1 dx 7 f x cos x dx 6 5 2 x dx 1 8 J x 2 dx x

5 24 Find the general indefinite integral 3x 4x 1 dx 5 6 5 2 x dx

Thinking 7 Differentiate the following without simplifying f x 4 cos x sin cosx

3 Differentiate each of the following Do not simplify a f x x cos 3x 9x b f x sinx tan x

1 Determine the derivative with respect to x Simplify fully a f x 12 4x c f x e6xsin 5x b y 1 5e x 2 d f x sin esinx

1 4 Verify by differentiation that the formula is correct 1 fln x dx x lnix x C 2 f tan x dx tan x x C 3 S 1 x 1 x dx 4 x a bx dx S 1 x 2 1562 3bx C 2a a bx 2 C

1 4 Verify by differentiation that the formula is correct 1 f In x dx x lnix x C 2 tan x dx tan x dx tan x x C

Follow the steps for graphing a rational function to graph the function P x x x 6 36 Determine the vertical asymptote s if one exists Select the correct choice below and if necessary fill in the answer box es to complete your choice A The function has one vertical asymptote Type an equation Use integers or fractions for any numbers in the equation B The function has two vertical asymptotes The leftmost asymptote is x 6 and the rightmost asymptote is x 6 Type equations Use integers or fractions for any numbers in the equations the middle asymptote is and the rightmost asymptote is OC The function has three vertical asymptotes The leftmost asymptote is Type equations Use integers or fractions for any numbers in the equations D The function has no vertical asymptote Determine the behavior of the graph on either side of any vertical asymptotes if one exists Select the correct choice below and if necessary fill in the answer box es to complete your E and on the other It approaches either co or co on both sides of the asymptote s at x OA It approaches o on one side of the asymptote s at x Type integers or simplified fractions Use a comma to separate answers as needed Type each answer only once OB It approaches either or on both sides of the asymptote s at x Type an integer or a simplified fraction Use a comma to separate answers as needed Type each answer only once and o on the other OC It approaches oo on one side of the asymptote s at x Type an integer or a simplified fraction Use a comma to separate answers as needed Type each answer only once D The function has no vertical asymptote

A company determined that the marginal cost C x of producing the xth unit of a product is given by C x x 4 Find the total cost function C assuming that C x is in dollars and that fixed costs are 6000 C x

Differentiate y xe 7x 8e x x8 Choose the correct answer below O A dy 7xe 7x e 7x 8e x 8x dx OB dy dx OC dy dx OD dy 7xe 7x e 7x 8e x 8x 7xe 7x e 7x 8e X 8x 7xe 7x e 7x 8e 8x dx

Suppose that a particle following the path c t t t 5t 0 flies off on a tangent at to 12 10 Compute the position of the particle at the time t Enter your answer in the vector form Use symbolic notation and fractions where needed

10 points Use the intermediate value theorem and the bisection method to approximate the real zero in the indicated in You only need to approximate to one decimal place f x 72 2x 5x 1 on 0 1

Solve the inequality f x 0 where f x x 1 x 2 by using the graph of the function 4 2 30 20 10 10 20 30 40 50 2 4 X C The solution set for f x 0 is Type your answer in interval notation

37 If r x y z and ro points x y z such that r ro 1 xo yo Zo describe the set of all

17 Find a unit vector with the same direction as 8i j 4k

13 16 Show that the equation represents a sphere and find its center and radius 13 x y 2 2x 4y 8z 15

36 Find the derivative of the function using the definition of derivative f x 5x 2 4x 3

3 Which of the points A 4 0 1 B 3 1 5 and C 2 4 6 is closest to the yz plane Which point lies in the xz plane

For the following exercises consider this scenario For each year t the number of trees in Forest A is represented by the function A t by the function B t 94 1 025 1 Which forest s population is growing at a faster rate 2 Which forest had a greater number of trees initially 3 Initially by how many trees did the larger forest exceed the smaller forest 4 Which forest will have a greater number of trees after 30 years 5 After 30 years by how many trees will the larger forest exceed the smaller forest Round to two decimal places 100 1 015 In the nearby Forest B the number of trees is represented 4

Use Newton s method with the specified initial approximation x to find x3 the third approximation to the root of the given equation Round your answer to four decimal place x5 x 7 0 X 1 x X3 3 2677

48 67 Find the four second order partial derivatives for f x y 5x y 4xy7 45 f 11

Let R x C x and P x be respectively the revenue cost and profit in dollars from the production and sale of x items If R x 7x and C x 0 005x2 2 6x 60 find each of the following a P x b R 200 C 200 and P 200 c R x C x and P x d R 200 C 200 and P 200

Explain why Newton s method doesn t work for finding the root of the equation x3 3x 9 0 if the initial approximation is chosen to be x 1 f x x 3x 9 f x in trying to Select by zero If x 1 then f x and the tangent line used for approximating x2 is Select Attempting to find x2

If 10 800 cm of material is available to make a box with a square base and an open top find the largest possible volume of the b cm3

Vrite an equation of a circle with diameter AB A 3 3 B 17 17 Choose the correct answer below A x 10 2 y 10 98 OC x 10 y 10 98 E x 10 2 y 10 3 OG x 10 y 10 98 OB x 10 y 10 OD x 10 y 10 OF x 3 y 3 1 OH x 17 2 y 17

Determine algebraically whether the given function is even odd or neither g x 3x 5 ONeither O Even Odd

HQ Four Quadrant Ordered Pairs B C 3 3 G E C Z 4 3 2 4 S Write the ordered pair for each given point 8 4 9 W 11 I 10 12 Z 2 2 2 1 1 2 4 4 5 8 Tell what point is located at each ordered pair 1 3 7 3 6 4 2 8 7 4 9 7 8 9 5 8 9 6 3 8 13 G 14 T Plot the following points on the coordinate grid 17 O 7 4 19 V 2 5 18 F 3 6 20 A 2 9 W X messa 21 Y 1 6 22 U 2 5 1 2 6 1 D 7 4 4 8 1 3 15 D 8 6 16 J 3 1 23 M 1 5 24 N 8 8

Find the slopes of the curve y y x at the two points shown here 4 coll Ay 0 Q The slope of the curve y y x at the point Type an exact answer using radicals as need

Find the critical points of the following function Use the Second Derivative Test to determine if possible whether each critical point corresponds to a local maximum local minimum or saddle point If the Second Derivative Test is inconclusive determine the behavior of the function at the critical points 2 2 f x y 4x 6xy 4x 1 What are the critical points Type an ordered pair Use a comma to separate answers as needed

12 5 Compute the directional derivative of the function g x y sin x 5x 4y at the point P 1 3 in the direction 5 13 13 The directional derivative is Type an exact answer using x as needed CO Be sure to use a unit vector for the direction vector

Compute the directional derivative of the function g x y sin x x 3y at the point P 2 3 in the direction The directional derivative is Type an exact answer using as needed 14 3 5 5 Be sure to use a unit vector for the direction vector

Find the Maclaurin series for the given function 14X 1 X What is the Maclaurin series of f x 80 k 0 4 1 X Type an expression using k and x as the variables

Use a graphing utility to approximate the solutions of the equation in the interval 0 2x Enter your answers as a comma separated list Round your answers to four decimal places sin x cos x X