Application of derivatives Questions and Answers

Use technology to find points and then graph the function y 2log 2 8 1 following the instructions below done plotting points Plot at least five points with integer coordinates that fit on the axes below Click a point to delete it

2 Determine the absolute extrema of the function on the closed interval given K f x 4x x 1 where x 0 4

8 A boat leaves a dock at 2 00 pm heading west at 15 km h Another boat heads south at 12 km h and reaches the same dock at 3 00 pm At what time were the boats closest to each other Round to 2 decimal places Interval not needed

4 1 Determine the absolute maximum and minimum values for the function below on the given interval K 1 5 S x x x 6x where 0 x 4 3 2 Minimum Value at x

Determine the absolute extrema of the function on the closed interval given K f x 4x x 1 where x 0 4

2 7 Finding where the derivative of a function equals zero will always produce the desired optimal value Is this statement true or false Explain your reasoning C

6 Explain why we cannot determine maximum or minimum values for the function f x X x 3 interval 5 x 5 Include in a sketch in your solution C over the

5 A cylindrical tank with no top is to be built from stainless steel with a copper bottom The tank is to have a volume of 5 m If the price of copper is five times the price of stainless steel what should be the dimensions of the tank so that the cost is a minimum Interval not needed A Vcylinder h SACylinder 2tr 2 rh

nalyze the polynomial function f x x x 5 using parts a through h below Determine the end behavior of the graph of the function he graph of f behaves like y for large values of x

Hiring nilai us X 2x J 1 x

4 Points DETAILS SPRECALC7COREQ 3 5 001 The polynomial P x 5x x 2 x 8 has degree It has zeros 0 2 and The zero 0 has multiplicity and the zero 2 has multiplicity

Given the graph of f x below find the graph of the derivative of f x Select the correct answer below 6 5 4 3 2 SH Y N 3 4 5 6

Find the open interval s of f x graph given below where f x is negative 6 Provide your answer below S 43 4 3 2 6 5 4 3 2 1 1 2 3 4 5 6 5 6 Give your answer in interval notation For example 2 4 U 6 00 141

5 and p Construct a Binomial probability distrubtion table using n 0 24 Next record the mean am standard deviation of the distribution below Where necessary round all numbers to four decimal places x P x

57 60 Use a graphing device to graph the polar curve Choose the parameter interval to make sure that you produce the entire curve STUT 0 57 r esin 2 cos 40 butterfly curve

2 Consider the series n 1 n 2 n 1 a x Use your computer to graph the functions f x 32 and g x 3 2 on the same set of axes You can use Desmos or your graphing calculator Set the scale on the horizontal axis so the x values go from 1 to 100 We are interested in comparing the functions for large values of x and when x is large both f x and g x are quite small Thus you should set the scale on the vertical axis to view the y values between 0 and 1 If you are using Desmos use the wrench button at the top right of your screen to set the vertical scale b Now use your computer to plot the function f x from part a together with h x You may have to experiment to find a range of y values that will enable you to see both functions on the same axes f x f x limx and limx h for the functions in parts a and b c Calculate above You can do this by hand or use the graphs of the quotients on desmos and see what happens as x goes to You don t have to show your work here but what are the limits d Look at the results of parts a through c and the statement of the LCT Should 1 you compare the series En 1 n 2 7 with n 1 7 2 or with n 1 Explain your choice after considering which term dominates the denominator for large values of n n 1n n e Does the series n 1 n 2 n converge Use the LCT and your knowledge of p series to justify your conclusion

Given the graph of f x below what is the open interval s over which f x is positive 6 5 4 3 H 1 2 3 4 5 6

Given the graph of f x below identify the graph of f x 6 5 4 3 2 1 6 5 4 3 1 2 3 4 31 H 1 2 3 4 5 6

question You can retry this question below A large company determines that in t months the number of units in the thousands that they will have 11t 3 produced is given by p t t 4 dollars Question Help Message instructor What will happen to the production in the long run

Evaluate the limit lim 3 tan 4x Enter a simplified multiple of or or x48

Question 12 Evaluate lim sin 9x o A

Evaluate the limit using L H pital s rule 7x lim X18 e5x

Question 4 lim x 0 Evaluate the limit using L Hospital s rule 4 5 X

The lesser til is a zentiplicity so the graph on OA o OA d Use a graphing utility to graph the function The graphs are shown in the viewing window Xmin B pants a through n Classes B Ay 1600 IOW e Approximate the turning points of the graph The turning point s of the graph is are 1 25 759 69 Type an ordered pair Round each coordinate to the nearest hundredth as needed Use a comma to separate answers as needed f Use the information in parts a through e to draw a complete graph of the function by hand Choose the correct graph below S X The gitaiti i is a ti miplicity the graph in Crosses f A axis al x 4 10 Xmax 10 Xscl 1 Ymin 1600 Yn 1600 Y 160 Choose the correct graph below max O C C 1 1600 O D D 57 Q

e Approximate the turning points of the graph The turning point s of the graph is are 3 16 34 33 5 16 6 90 Type an ordered pair Round each coordinate to the nearest hundredth as needed Use a comma to separate answers as needed f Use the information in parts a through e to draw a complete graph of the function by hand Choose the correct graph below OA wer parts a through h B Ay g From the graph find the range of the polynomial function The range of the polynomial function f is 0 00 Type your answer in interval notation Round to the nearest hundredth as needed h Use the graph to determine where the function is increasing and where it is decreasing The function is increasing on the interval s pooded Use a comma to separate answers as needed D

3 If the price charged for a candy bar is p x cents then x thousand candy bars will be sold in a certain city where p x 149 2 How many candy bars must be sold to maximize revenue A 1937 thousand candy bars C 1937 candy bars B 3874 candy bars D 3874 thousand candy bars

Analyze the polynomial function f x 3 x 3 x 4 using parts a through h below The graph of f behaves like y 3x for large values of x b Find the x and y intercepts of the graph of the function The x intercept s is are 3 4 Simplify your answer Type an integer or a fraction Use a comma to separate answers as needed Type each answer only once The y intercept s is are 576 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 3 4 Simplify your answer Type an integer or a fraction Use a comma to separate answers as needed Type each answer only once Y min 1600 Y max The lesser zero is a zero of multiplicity 1 so the graph of f crosses the x axis at X 3 The greater zero is a zero of multiplicity 3 so the graph of f crosses the x axis at x 4 d Use a graphing utility to graph the function The graphs are shown in the viewing window Xmin 10 Xmax 10 Xscl 1 1600 Yscl 160 Choose the correct graph below OA www C B O C OU O D

ind the general form of the quadratic function with the vertex 3 2 and a point on the graph 5 6 f x help formulas

Find the general form of the quadratic function that has a vertex of 2 2 and a point on the graph 5 51 f x help formulas

Rewrite the quadratic function f x x 8x 3 in standard form f x a x h k and give the vertex help numbers f x Vertex Ex Quadratic Function Reviewerview Use f x 2x 45x 3 for the following questions f x ax bx 1 Does the parabola open up or down The parabola opens down 2 Determine the equation of the axis of symmetry b X 2a the vertex Xa 3 Determine the co H Q x 4 Determine the x and y intercepts x ints set f x 0 y int set ka0 2 0 3 0 help points 6 2 8 S 2 0 2x 5x 3 0 27 5x 3 OF 2x 12x 3 2x 1 0 or X 3 0 x 3 G

For the quadratic function f x x 4x 5 find the vertex the axis of symmetry and the x intercept s Express the vertex as a coordinate pair If there is more than one intercept enter as a comma separated list of points help points vertex axis of symmetry x intercept s help numbers help points

Determine whether f x 4x 16x 4 has a minimum or maximum value and find the value of the minimum or maximum Also find the axis of symmetry The y value of the is The axis of symmetry for f x is given by help numbers help numbers

Determine the domain and range of f x 7 x 5 3 Enter the solution in interval notation help intervals Domain Range help intervals

C f x x 0 16 Graph the function the secant line through the endpoints and the tangent line at c f c y y 5 15 10 5 y 15 10 1 2 3 5 3 5 10 15 1 separated list If an answer does not exist enter DNE

13 3 Analyze the polynomial function f x x x 7x 21 Answer parts a through h 24 8 8 3 4 8 8 e Approximate the turning points of the graph The turning point s of the graph is are 3 16 34 33 5 16 6 90 Type an ordered pair Round each coordinate to the nearest hundredth as needed Use a comma to separate answers as needed f Use the information in parts a through e to draw a complete graph of the function by hand Choose the correct graph below OA www B 100 Ay O C O D

For the polynomial function f x 8x x 2 answer the following questions a List each real zero and its multiplicity b Determine whether the graph crosses or touches the x axis at each x intercept c Determine the maximum number of turning points on the graph d Determine the end behavior that is find the power function that the graph of f resembles for large values of x b Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The graph crosses the x axis at Type an exact answer using radicals as needed Type an integer or a simplified fraction Use a comma to separate answers as needed OB The graph touches the x axis at Type an exact answer using radicals as needed Type an integer or a simplified fraction Use a comma to separate answers as needed C The graph touches the x axis at 0 and crosses at 2 2 Type integers or simplified fractions Use a comma to separate answers as needed OD The graph neither crosses nor touches the x axis c The maximum number of turning points on the graph is Tyng a whole number

8 A town with a population of 100 people grows to a population of 200 in five years Assume the dP population P grows exponentially according to the equation kP Find the relative growth rate dt k

2 Compute the derivative for the function y x e dy dx Note you do not need to simplify your answer

5 When 2x xy 4y 2 express dy in terms of x and y using implicit differentiation dx

Analyze the polynomial function 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 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 A Answer parts a through h The least zero is of multiplicity 1 so the graph of f crosses the x axis at x graph off 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 5 B 7 The middle zero is of multiplicity 1 so the graph of f crosses the x axis at x 3 The greatest zero is of mult 10 Xmax 10 Xscl 1 Ymin 50 Ymax 50 Y sci 5 Choose the correct graph below O C Q O D o O 5

The radius of a sphere is increasing at a rate of 3 mm s How fast is the volume increasing when the diameter is 80 mm mm s

R If two resistors with resistances R and R are connected in parallel as in the figure below then the total resistance R measured in ohms 22 is given by 1 1 2 22 R If R and R are increasing at rates of 0 3 2 s and 0 2 2 s respectively how fast is R changing when R 100 2 and R 110 2 Round your answer to three decimal places 22 s www MY NOTES R PRACTIC

h x 2x 15x 144x 1 Given the function above use the First Derivative Test to find the local extrema Select the correct answer below There is a local maximum at x 8 O There is a local maximum at x 3 O There is a local minimum at x 8 and a local maximum at x 3 There is a local maximum at x 8 and a local minimum at x 3 There is a local minimum at x 3

The line y 5 is a horizontal asymptote to the graph of which of the following functi sin 5x y B y 5x C y 5x D 1 x 1 x 5 E y 20x 1 4x

Graph of 18 The graph of a function is shown above Which of the following limits does not exist E lim f x A lim f x B im f x C lim f x D lim f x 19 Let be the function defined by f x 3x 8 5 4x Which of the following is a horizontal asymptote to 2x 1 2 the graph of f A y 6 B y 3 0 y 12 2 D y 0 E y 2

B 2 16 C 3 D nonexistent If f is a continuous function such that f 2 6 which of the following statements must be A lim 2x 3 B lim 2x 12 f x f 2 6 C lim D lim x 36 E lim f x 36 7 For which of the following pairs of functions f and g is lim 818 A f x x 2x and g x x ln x B f x 3x and g x x C f x 3 and g x x D f x 3e x and g x E f x In 3x and g x In 2x and g x 2e x f x infinite g x

1 Is the function f x continuous at x 27 Give a reason for your answer t hrs v t miy hr 1 2 4 5 7 3 8 6 10 3 2 A bug walks back and forth along a straight track The bug s velocity measured in miles per hour i ontinuous with selected values of the velocity shown in the table above time for 4 7 such that the velocity of the bug equals 0 Explain your reasoning

et x 0 Find y using implicit differentiation if xy y 1 0

Here is the graph of a quadratic function f 5 0 12 10 5 0 2 0 5 4 4 Of x x 4 4 5 6 0 Select all equations that could define the function f Of x x 4 4 Of x x 8x 12 Of x x 2 x 6 Of x x 2 x 6 f x x 8x 12 10

Prove the identity tan x 1 tan x 3 1 3 tan x 1 3 tan x tan x tan F 3 tan x 3 1 3 tan x 3 tan x 1 3 tan x X