Application of derivatives Questions and Answers

X Follow the steps for graphing a rational function to graph the function G x 4 If needed first write the given function as a single rational expression Then factor the numerator and denominator of G x Select the correct choice below and if necessary fill in the answer box within your choice 1x x 2 x 2 OB G x cannot be factored further What is the domain of G x Select the correct choice below and if necessary fill in the answer box within your choice Type an integer or a simplified fraction A G x Type your answer in factored form Do not simplify OA xx OB xx Type an integer or a simplified fraction Use a comma to separate answers as needed OC x x Type an integer or a simplified fraction OD The domain is the set of all real numbers

Follow the steps for graphing a rational function to graph the function R x ww 5x 5 8x 32 If needed first write the given function as a single rational expression Then factor the numerator and denominator of R x Select the correct choice below and if necessary fill in the answer box to complete your choice A R x 5 x 1 8 x 4 OB R x cannot be factored further What is the domain of R x Select the correct choice below and if necessary fill in the answer box to complete your choice Type your answer in factored form Do not simplify OA X X Type an integer or a simplified fraction Use a comma to separate answers as needed OB x x 2 Type an integer or a simplified fraction OC xx Type an integer or a simplified fraction OD The domain is the set of all real numbers

3 Evaluate the following limits cos x a lim sin x 0 COS X X 1 3 1 x1 x 1 b lim

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 places x5 x 7 0 x 1 X X3 3 2677

99 Suppose 7 a 6 3 2 b 8 c 16 d 2 2 e 6 3 2 12 7 4 and and let g x f x cosx Find TU

6 The slope of the tangent line to the graph of the exponential function y 9 at the point 0 1 is lim 9x Estimate the slope to three decimal places X 0 a 2 485 b 2 303 c 2 565 d 2 197 e 2 079 X

A car was valued at 38 000 in the will it be worth by 2017 year 2007 By 2013 the value had depreciated to 11 000 If the car s value continues to drop by the same percentage w help numbers

Provide a detailed sketch with analysis of the function f x x on the domain 0 Your sketch must include the following details all of which must be justified any local and any global extrema of f limo f x limx f x concavity of f

The fox population in a certain region has an annual growth rate of 4 per year In the year 2012 there were 27790 foxes counted in the are Write a function f t that models the number of foxes in the population at t years after 2012 Assume a continuous growth rate f t help formulas What is the fox population predicted to be in 2020 help numbers

3 Compute lim x arctanx 1 x T 2 Justify your answer

130t 7t 100 Click here to access the Desmos Graphing Calculator and use it to approximate the time when the concentration is highest help numbers The concentration C of a drug in a patient s bloodstream t hours after injection is given by C t t GI

Consider the equation y 5 82e 0 0349t which is of the form y Aert to fill in the following blanks The r value for the equation is Therefore the type of change represented is because

For the function f x 6 5 find f 3 Round to five decimal places if necessary help numbers f 3

Find the volume of the following solid The tetrahedron bounded by the coordinate planes x 0 y 0 2 0 and the plane 5x 3y z 15 0 The volume is cubic units Simplify your answer Type an integer or a fraction C

Question 13 Given P x x 81 Write Pin factored form as a product of linear factors Be sure to write the full equation including P x

estlo 8 Divide and simplify 101 23i 7 3i

Simplify

Question 6 Perform the indicated operation simplify Express the answer as a complex number 8 10i 5 2i

3 Graph the piecewise function given by 4 10 x 5 5 x 2 x 2 g x X 2 10 8 6 V 4 2 10 8 6 4 2 0 2 4 6 8 10 2 4 6 8 10

poster is to have an area of 210 in2 with 1 inch margins at the bottom and sides and a 2 inch margin at the top Find the exact dimensions that will give the largest printed area vidth 3 35 X in height 2 35 X in

0 Describe the locus of points in a plane consisting of the centers of all circles with a radius of 2 cm that are tangent to the given line m Hint Do not assume that the locus has only one part

Solve the inequality Enter your answer using interval notation x 7 x 3 x 1 0

Solve the inequality Enter your answer using interval notation 6x3x 9 54x

Name Instructions Use algebraic techniques to solve problems unless otherwise instructed Show the calculations leading to your answers except basic arithmetic Simplify and circle your final answers 1 A certain breed of mouse with an initial population of 30 was introduced onto a small island Scientists estimate that the mouse population doubles every year a 4 pts Find a function N that models the number of mice after t years b 4 pts Use the function to estimate the mouse population after 4 years and 6 years assuming that none have died

3 3 points Evaluate the following integral some need to be changed to other coordinates a b c 2y px y 6 1 1 y 0 IM LS 9 x 9 x y 1 x y ry dzdxdy 1 x y x y dzdydx x z y z z dzdxdy

Solve the given equation Enter your answers as a comma separated list Let k be any integer Round terms to three decimal places where appropriate If there is no solution enter NO SOLUTION 2 sin 0 cos 0 1 8

Problem 5 a Find the value of c that satisfies the conclusion of the Mean Value Theorem for the function f x lnx on the interval 2 16 Problem 5 a Problem 5 b b If f 1 4 and f x 9 for 1 x 9 then according to the Mean Value Theorem what is the smallest possible value for f 9 c Let f be the function whose graph is shown below 3 N 3 y 2 1 0 0 1 2 4 X Estimate the two values of c that satisfy the conclusion of the Mean Value Theorem on the interval 2 7 5 6 7 8 Enter your answer symbolically as in these examples

An equation is given Enter your answers as a comma separated list Let k be any integer Round terms to three decimal places where appropriate If there is no solution enter NO SOLUTION 3 tan 30 1 0 a Find all solutions of the equation 0 b Find the solutions in the interval 0 2 8

Solve the given equation Enter your answers as a comma separated list Let k be any integer Round terms to three decimal places where appropriate If there is no solution enter NO SOLUTION 2 sin 8 cos 8 1 8

Problem 9 Suppose that the derivative of a function f is given by f x x x 2 6 x 3 5 x 4 Find the x coordinates of the local maximum points of f Problem 9 A 0 and 2 only G 2 and 3 only D B 3 and 4 only C 0 and 3 only D 2 and 4 only E 3 only F 0 and 4 only H 2 only

Problem 10 Suppose that the second derivative of a function f is given by f x x x 2 7 x 3 4 Find the x coordinates of the inflection points of f if any A 2 only B 2 and 3 only C 0 only D 0 and 2 only E 0 and 3 only F none G 3 o H 0 2 and 3

6 Let f x 1 e x On what interval is f increasing A In 6 B 0 In 6 C In 6 D In 5 0 E In 5 F In 5 0 G In 6 H ln 5

1 Maisy is an employee at a shoe store She receives a 20 discount on all items including sales items For Black Friday all boots at the store are discounted 40 Maisy grabs a pair of boots that are regularly sold for 80 and takes them to the register where her employee discount is applied to the sale price How much did Maisy have to pay for her boots

Pierce Manufacturing determines that the daily revenue in dollars from the sale of x lawn chairs is R x 0 004x 0 a What is the current daily revenue b How much would revenue increase if 75 lawn chairs were sold each day 0 05x 0 5x Currently Pierce sells 70 lawn chairs daily c What is the marginal revenue when 70 lawn chairs are sold daily d Use the answer from part c to estimate R 71 R 72 and R 73

The function f x y z xyz 1 2 has an absolute maximum value and absolute minimum value subject to the constraint x y z 4 with x 20 y 20 and z 20 Use Lagrange multipli to find these values The absolute maximum value is 4 4 3 3 The absolute minimum value is 0 GEER

A company s marginal cost function is MC x x 1 2 5 where x is the number of units If fixed costs are 23 250 find t company s cost function C x

8 Identify which of the following trig functions represents that of a sin cos tan cot csc and sec function f 4 2 A U 44

A model used for the yield Y of an agricultural crop as a function of the nitrogen level N in the soil measured in appropriate units is KN 9 Y where k is a positive constant What nitrogen level gives the best yield N

3a Give a definition for the piecewise function shown below 5 3b Identify the discontinuities of the function 0

Sketch the graph of a function f that is continuous on 1 5 and has the given properties Absolute maximum at 5 absolute minimum at 2 local maximum at 3 local minima at 2 and 4 3 3 25 O s 0 5 3 2 5 2 15 1 2 3 4 5 6

Find the absolute minimum and absolute maximum values of f on the given interval f x 0 6 x 2 x 4 absolute minimum value absolute maximum value

10 Solve for side length a in the triangle below Round your answer to one decimal place X 35

Find the critical numbers of the function Enter your answers as a comma separated list Use n to denote any arbitrary integer values If an answer does not exist enter DNE f 0 4 cos 0 2 sin 0 x

Find the critical numbers of the function Enter your answers as a comma separated list If an answer do not exist enter DNE f x 8x 12x 48x H

Explain the difference between an absolute minimum and a local minimum There is no difference O A function f has an absolute minimum at x cif f c is the largest function value on the entire domain of f whereas f has a local minimum at c if f c is the largest function value when is nea C A function f has an absolute minimum at x c if f c is the smallest function value when a is near c whereas f has a local minimum at cif f c is the smallest function value on the entire domain of f A function f has an absolute minimum ata cif f c is the largest function value when z is near c whereas f has a local minimum at cif f c is the largest function value on the entire domain of f OA function f has an absolute minimum at x cif f c is the smallest function value on the entire domain of f whereas f has a local minimum at cif f c is the smallest function value when z is

Find the slopes of the curve y y2 x at the two points shown here Q Q 194 3 2 3 3 The slope of the curve y y x at the point Type an exact answer using radicals as needed The slope of the curve y y x at the point Type an exact answer using radicals as needed 3 4 is is

nd the slopes of the curve y4 y2 x at the two points shown here Ay 3 1 2 oo 3 4 The slope of the curve y4 y2 x at the point Type an exact answer using radicals as needed The slope of the curve y4 y2 x at the point Type an exact answer using radicals as needed 3 4 1 2 is 3 3 2 is

Find all solutions of the equation in the interval 0 360 Enter your answers as a comma separated list cos x 1

Find the area shared by the circle r2 4 and the cardioid r 4 1 cos 0 The area shared by the circle and the cardioid is Type an exact answer using as needed

Find the length of the spiral r 02 0 0 45 The length of the spiral is Simplify your answer