Differentiation Questions and Answers

60 50 40 30 20 10 10 Type answer below 20 61718 1 30 40 9 50 10 1 11 12 113 114 115 1 16 Maximum output for TC of 400 000 60 a State the least cost combination of inputs for this firm TPP 4 500 TC 400 000 Labour L www economichep org TPP 4 000 TPP 3 500 2 marks b State the production level that will be produced using the least cost method 1 mark 17 181 19

2 1 a The domain of f x 2 4 b The range of f x 1 3 The graph of y f x given above starts at x 2 and ends at x 4 Use the graph of y f x to identify 2 47 c f 0 1 2 d For what values of x does f x 1 Doesn t wom 2 2 a b 1 3 C d M

Find the derivative of the following a f x x 3x 2 x 5 b f x C d e f f x 132 f x 10x f x 2 36 y x f x 3 2x 5x 4 2x 3 4 3 7x 4x 3x 1 2x 1 2 9 f x x 3 1 2x g www 2 11x 12x 16

se implicit differentiation to find dy dx 15 x3 3x2y y3 8 x2 3xy x y2 A B x2 2xy x2 y2 x2 2xy C x2 y2 D x2 32 x2 y

According to a study a vehicle s fuel economy in miles per gallon mpg decreases rapidly for speeds over 60 mph a Estimate the speed at which the absolute maximum gasoline mileage is obtained b Estimate the speed at which the absolute minimum gasoline mileage is obtained c What is the mileage obtained at 70 mph C Fuel economy mpg 36 30 24 18 12 6 55 65 75 85 95

For the graph shown identify a the point s of inflection and b the intervals where the function is concave up or concave down a The point s of inflection is are Type an ordered pair Use a comma to separate answers as needed b Identify the intervals where the function is concave up or concave down Z 10 O N The function is concave up on and concave down on Simplify your answers Type your answers in interval notation Use a comma to separate answers as needed

Find the intercepts of f x 2 x 5 x 5 x 3 Enter intercepts as points If you have more than one point enter them as a comma separated list help points y intercept x intercept s help points

87 A manufacturer determines the relationship between the demand and price for an item is given by 2x 6xp 30p 125 000 where x is the number of items demanded each month at a price of p dollars each If the price per item is increasing at a rate of 3 50 per month find the rate of change of the demand with respect to time when the price is 50 per item Interpret your answer Round to the nearest integer if necessary

x u v u 3 x ex 3 6 ubtex xe U ex litur e exx e xe P 3 x V 9 6e p L abe 2ex 6e7 2e 51 J 2 4 1 12x 6e X X 5 be

Consider the function p x Select the correct answer below 20 V 1200 32 554 20 V 1200 32 16x 125x What are the critical numbers 5 20 1200 32 20 1200 32

29 Find the second derivative of g x e ln 4x

For the following exercises use logarithmic differentiation to find dy dx 346 y xVx X

341 f x ln 5x 7

3 The function f x 2x 33x 180x 3 has one local minimum and one local maximum Use derivatives to algebraically find the answer to the question This function has a local minimum at x with y value and a local maximum at x with y value

4 1 Determine the absolute maximum and minimum values for the function below on the given interval K f x x 3x 6 x 6x where 0 x 4 2 Minimum Value at x

9 The difference between two numbers is k Show that their product is at least 0 25k Interval not needed

9 The difference between two numbers is k Show that their product is at least 0 25k Interval not needed

4 A car rental agency has 200 cars The owner finds that at a price of 36 per day he can rent all the cars For each 2 increase in price the demand is less and 5 fewer cars are rented What price will maximize the total revenue A Interval

6 of the swimming area is a straight sandy beach and the rope will form the other three sides Determine the dimensions of the rectangle that produces the maximum swimming area Determine the dimensions that would produce the largest swimming area A Interval

3 ex f x nv uv 3 e x ex 3 6 ubtex xe u s ex livtur e e x e xe V 3 ex v 9 62 12 va 6e 2e 31 J 2 Cafilah derivative f x 6x x ex 3x

Find all zeros of the polynomial Enter your answers as a comma separated list Enter all answers including repetitions P x 2x 9x 16x 15 X 5 2 27 1 2i 1 X Your answer cannot be understood or graded More Information

51 54 Find the points on the given curve where the tangent line is horizontal or vertical 51 r 3 cos 0 53 r 1 cos 0 52 r e 54 r sin 20

The graph of function f x is shown below At which value of x is the derivative of the function equal Select the correct answer below Ox 4 0 3 O x 2 Ox 1 0x 0 HIN 6 5 4 3 2 1 3 2 1 3 1 2

47 50 Find the slope of the tangent line to the given polar curve at the point specified by the value of 0 47 r 2 sin 0 0 6 48 r 2 sin 0 0 3

HIV JR Simplify your answer Type an integer or a fraction Use a comma to separate answers as needed Type each answer only once c Determine the real zeros of the function and their multiplicity Use this information to determine whether the graph crosses or touches the x axis at each x intercept The real zero s of f is are 0 7 Simplify your answer Type an integer or a fraction Use a comma to separate answers as needed Type each answer only once n below The lesser zero of the function is of multiplicity 2 so the graph off touches the x axis at x 0 The greater zero of the function is of multiplicity 1 so the graph of f crosses the x axis at x 7 d Use a graphing utility to graph the function The graphs are shown in the viewing window Xmin 20 Xmax 20 Xscl 2 Y min 100 Ymax 100 Yscl 10 Choose the correct graph below OA B Q O C D 10

Find the dimensions of a rectangular corral split into 2 pens of the same size producing the greatest possible enclosed area given 100 feet of fencing help numbers Width Length help numbers

Quest no Evaluate the limit using L H pital s rule X e 2x 1 6x lim x 0

Find the domain and range for the quadratic function with a vertex 10 0 if the graph opens down help intervals Domain M Range help intervals

Let f x x 3 2 Find all values of c in 1 4 such that f 4 f 1 f c 4 1 Enter your answers as a comma separated list If an answer does not exist enter DNE C Based off of this information what conclusions can be made about the Mean Value Theorem O This contradicts the Mean Value Theorem since f satisfies the hypotheses on the given interval but there does not exist any c on 1 4 such that f c This does not contradict the Mean Value Theorem since f is not continuous at x 3 f 4 f 1 This does not contradict the Mean Value Theorem since f is continuous on 1 4 and there exists a c on 1 4 such that f c This contradicts the Mean Value Theorem since there exists a c on 1 4 such that f c f 4 f 1 but f is not continuous at x 3 4 1 Nothing can be concluded f 4 f 1 4 1

Follow the instructions in each problem Show supporting work not just a final answer to receive cre 1 Compute the second derivative d dx2 218

7 Compute the derivative of f x 2 en r

4 Find the derivative of f x cos x Note you do not need to simplify your answer

6 Compute the derivative of y ln x 5

Each side of a square is increasing at a rate of 2 cm s At what rate is the area of the square increasing when the area of the square is 81 cm cm s

31 02 03 04 1 5 25 Lt fbe the function given by f x 1 2 2 x 20 D nonexistent the line yb is a borineal asympaote to the graph Which of the following is true has a pimp discontinuity at 2 2 D has a discontinuity due to a vertical asymptote at x 2

A 2 19 2mx for x 2 i 11 Let be the function defined above where e is a constant If f is continuous at x 1 what is the value of 7 A 2 C S continuous A 3 and 1 only 3 1 and 2 1 only 1 and 2 only only B 3 12 Let f be the function defined above where e is a constant For what value of c if any is f continuous at x 2 E There is no such value of c B 7 D 9 C 9 2 sin x f x 1x cx 18 D 4 4 3 Let f be the function given by f x x 2 x 1 for x 2 for x 2 x 2 x 3 For which of the following values of x is f not

8 The graph below shows the first derivative of a function y f x Select a possible graph of f that passes through the point P A C f M B D 4 X 8

1 From a thin piece of cardboard 30 in by 30 in square corners are cut out so that the sides can be folded up to make a box What dimensions will yield a box of maximum volume What is the maximum volume Round to the nearest tenth if necessary A 20 in x 20 in x 10 in 4000 in 3 C 10 in x 10 in x 10 in 1000 in 3 B 20 in x 20 in x 5 in 2000 in3 D 15 in 15 in 7 5 in 1687 5 in3 M

Question 3 20 points 3 Find the derivatives of the given functions a 2x 1 x y ln b In 2x In 1 x 3x y e x cos 4x

1 Find the derivatives of the given functions a y 2 sin 2x 1 b y 3 cos 5x 2

Verify the identity tan u sin u tan u sin u sin u tan u sin u 11 E sin u cos u sin u cos u cos u sin u cos u 1 tan u sin u 1 cos u cos u

a If a is a zero of the polynomial P x then b If a is a zero of multiplicity m of the polynomial P x then must be a factor of P x must be a factor of P x when we factor P completel

77 x Analyze the polynomial function f x The graph of f behaves like y 32 7x 7x 21 Answer parts a through h www for large values of x b Find the x and y intercepts of the graph of the function The x intercept s is are 7 3 7 Simplify your answer Type an integer or a fraction Use a comma to separate answers as needed Type each answer only once the x axis at x The y intercept s is are 21 Simplify your answer Type an integer or a fraction Use a comma to separate answers as needed Type each answer only once c Determine the zeros of the function and their multiplicity Use this information to determine whether the graph crosses or touches the x axis at each x intercept The least zero is of multiplicity so the graph of f so the graph of f the x axis at x The real zero s of f is are 7 3 7 Simplify your answer Type an integer or a fraction Use a comma to separate answers as needed Type each answer only once Sa the x axis at x The greatest zero is of multiplicity so the graph of f The middle zero is of multiplicity

Concavity and points of inflection Answer the following questions using the function f x e The first derivative of the given function is The second derivative of the given function is x 0 f has points of inflection that occur at x NOTE If you believe the answer is none type none Otherwise state the values as a comma separated list if needec f is concave down on ar and concave up on NOTE State the open intervals of concavity as a comma separated list if needed

Bonus 5 points Write a function such that f x f x 0 e

5 3 points 1 5 points each Determine the derivative of each function a f t 24 e 7 b g x e 5 67

dw dt t Use the Chain Rule to find where w cos 8x sin 2y x and y t 4 aw ax Type an expression using x and y as the variables

Prove the identity sin x 7 sin x Use the Subtraction Formula for Sine and then simplify sin x sin x sin x It II cos x sin 77 1 cos x 0

Prove the identity tan x 7 tan x Use the Subtraction Formula for Tangent and then simplify tan x tan tan x Tt tan x 1 1 tan x 0 tan x

Prove the identity sin x cos x 2 Use the Subtraction Formula for Sine and then simplify sin x 2 sin x cos cos x sin x 0 cos x