Differentiation Questions and Answers

Find the exact value of the trigonometric expression when sin u sin u v 12 and cos v 13 4 5 Both u and v are in Quadrant II

Find the exact values of the sine cosine and tangent of the angle 5TT 12 sin 512 5T Cos tan 5TT 12 5TT 12 11 11 11

Evaluate integral 16x 20 2x 5 fox

Question 4 Find all zeros of the function f x 3x 24x 51x30 Enter the zeros separated by commas Question Help Video Message instructor

A cylinder shaped can needs to be constructed to hold 200 cubic centimeters of soup The material for the sides of the can costs 0 04 cents per square centimeter The material for the top and bottom of the can need to be thicker and costs 0 06 cents per square centimeter Find the dimensions for the can that will minimize production cost Helpful information h height of can r radius of can Volume of a cylinder V h Area of the sides A 2 rh Area of the top bottom A To minimize the cost of the can Radius of the can Height of the can Minimum cost cents

r 30 O A 20 20 20 OB 20 20 20 O C 20 20 20 O D 20 20 20

Identify the symmetries of the curve below Then sketch the curve r sin 2 Identify the symmetries of the curve Select all that apply A The curve is symmetric about the y axis B The curve is symmetric about the origin C The curve is symmetric about the x axis

Note If you are asked for value s give your answer as a single value e g 17 or a comma separated list e g 1 2 3 or NONE If you are asked for an interval give your answer in interval notation Type INF for oo and INF for o Consider the function f x e 2 4 a List the location the x values of all the critical points of f x b Use interval notation to indicate where f x is increasing Increasing for in c Use interval notation to indicate where f x is decreasing Decreasing for a in d List the x values of all local maxima of f x values of local maxima e List the x values of all local mimima of f x values of local minima f List the x values of all the inflection points of f values of inflection points g Use interval notation to indicate where f x is concave up Concave up for a in h Use interval notation to indicate where f x is concave down BAY

Note If you are asked for value s give your answer as a single value e g 17 or a comma separated list e g 1 2 3 or NONE If you are asked for an interval give your answer in interval notation pe INF for oo and INF for 0 Consider the function f x x 4x 18x 15 a List the locations the x values of the critical points of f x b List the x values of all local maxima of f x values of local maxima c List the x values of all local mimima of f x values of local minima f List the x values of all the inflection points of f values of inflection points g Use interval notation to indicate where f x is concave up Concave up for a in h Use interval notation to indicate where f x is concave down Concave down for in

Note If you are asked for value s give your answer as a single value e g 17 or a comma separated list e g 1 2 3 or NONE If you are asked for an interval give your answer in interval notation Type INF for oo and INF for o Consider the function f x x 4 5x 12x 1 and its derivatives f x 3x 9x 12 f x 6x 9 a Determine the x intervals on which f is concave up and concave down f is concave up on f is concave down on b What are the x y coordinates of the inflection points of f Enter your answer as an x y coordinate e g 91 13 and separate multiple answers by commas

Suppose that f x x 3x a List the x values of all critical points of f x 3 Use interval notation to indicate where f x is increasing A function is increasing if larger x values give larger y values Don t include points where the function changes from increasing to decreasing or from decreasing to increasing Increasing for a in C Use interval notation to indicate where f x is decreasing A function is decreasing if larger x values give smaller y values Don t include points where the function changes from increasing to decreasing or from decreasing to increasing Decreasing for a in D List the x values of all local maxima of f x values of local maxima E List the x values of all local minima of f x x values of local minima F Use interval notation to indicate where f x is concave up Concave up for a in G Use interval notation to indicate where f x is concave down Concave down for a in H List the revalues of all the inflection points of f

8 The profit a company makes by manufacturing and selling a thromdimbulators is given in dollars by p x 0 1 100a 2000 Current production levels are a 300 units a Find the linearization L a to p x at 300 b Use the linearization to estimate the new profit obtained if production can be raised from 800 to 10 units

QUESTION 2 2 points Information on functions f z g z f z and g x for some z is given in the table Let h r gof x I g 1 Find A 2 and h 2 1 2 4 3 3 2 4 3 1 2 3 5 2 4 3 4 3 1 1 4

Find the slopes of the curve 4 y y x at the two points shown here 43 4 3 Ay 3 4 The slope of the curve y y x at the point 3 4 is 2 Type an exact answer using radicals as needed

At noon ship A is 50km west of ship B Ship A is sailing south at 25km h and ship B is sailing north at 30km h How fast is the distance between the ships changing at 4 00pm The distance between the ships is changing at a rate of kilometers per hour

If f is the focal length of a convex lens and an object is placed at a distance p from the lens then its image will be at a distance q from the lens where f p and q are related by the lens equation 1 1 1 f p 9 What is the rate of change of p with respect to q if q 5 m and f 6 m Make sure you have the correct sign for the rate m s

The following is a graph of the function f x e 5x Which of the following is the graph of f x A B Click on a graph to enlarge it 48 C

A company plans to manufacture a rectangular container with a square base an open top and a volume of 214 cm The cost of the material for the base is 0 3 cents per square centimeter and the cost of the material for the sides is 0 5 cents per square centimeter Determine the dimensions of the container that will minimize the cost of manufacturing it What is the minimum cost Round any calculations and the final answer to the nearest hundredth if necessary

You and a friend both leave the same restaurant to drive home You are heading directly west at 30 miles per hour and he or she is heading directly south at 40 miles per hour After half an hour how fast in mph is the distance between you changing Do not include units in your answer

Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit Assume that revenue R x and cost C x of producing x units are in dollam R x 60x 0 1x C x 4x 15 In order to yield the maximum profit of units must be produced and sold Simplify your answers Round to the nearest cent as needed COLD

Graph the tangent line to the function defined implicitly by the equation below at the point 1 2 3x x 2 2y y The black dot has already been placed at the desired point on the curve The green dot should be placed at another point along the tangent line Provide your answer below 10 14 10 0 10 10 5 2

Find an equation for the line tangent to the curve at the point defined by the given value of t Also find the value of d y 2 dx at this point x 16 sint y 4 cos t t The equation TU represents the line tangent to the curve at t Type an exact answer using radicals as needed T 4

A car rental company charges its customers p dollars per day to rent a car where 35 p 175 The number of cars rented per day can be modeled by the linear function n p 700 4p How much should the company charge each customer per day to maximize revenue Do not include units or a dollar sign in your answer Provide your answer below

Question Find the equation of the tangent line at the point 1 2 to the curve defined implicitly below x 3x 6 2y 3y

Question Find the equation of the tangent line at the point 1 2 to the curve defined implicitly below x x y 3

GIA STATE The administration conducted a survey to determine the proportion of students wh ride a bike to campus Of the 128 students surveyed 6 ride a bike to campus Which of the following is a reason the administration should not calculate a confidence interval to estimate the proportion of all students who ride a bike to campus Check all that apply The sample needs to be random but we don t know if it is The actual count of bike riders is too small The actual count of those who do not ride a bike to campus is too small

6 Let be a function that is differentiable on the open interval 1 10 If f 2 5 f 5 5 and f 9 5 which of the following must be true I f has at least 2 zeros II The graph of f has at least one horizontal tangent III For some c 2 e 5 f c S A None B I only C I II only D I III only E I II III

3 Dudley is dropped out of a plane and falls vertically toward the earth Dudley s acceleration is the sum of two components a downward acceleration due to gravity equal to g 9 8 kg m s an acceleration in the opposite direction to Dudley s current velocity and proportional to this velocity a Write a differential equation that describes Dudley s vertical velocity as a function of time Choose downward as the positive direction There will be an as yet undetermined constant of proportionality involved call it k b Compute an explicit formula for Dudley s velocity as a function of time assuming that i Dudley will approach a maximum vertical speed of 65 m s and ii that Dudley is shoved out of the plane with an initial downward velocity of 5 m s

Consider the function f x 6x 4x 4 0 x 10 Use the derivatives to algebrically answer the question The absolute maximum of f x on the given interval is at x and the absolute minimum of f x on the given interval is at

Consider the function f x x 72x 2 5 x 13 Use the derivatives to algebrically answer the question This function has an absolute minimum value equal to and an absolute maximum value equal to

4 Al 3 the function given by f x z if x 3 4 2x 1 if x 3 2 is A undefined B continuous but not differentiable S C differentiable but not continuous D neither continuous nor differentiable E both continuous and differentiable lim f x flam Jim 4 78

Graphing Directions Answer all three parts Consider the function f x graphed to the right f x 2 x 3 4 Part A Record all of the transformations of f x from the parent function g x x 10 Domain 0 Range 4 2 2 4 0 10 46 Partner B Determine the domain and range of the function Record your answer using set notation 10 12 14 16 18 Partner C For what interval s on the domain is the function increasing or decreasing How d you know Record your answer using interval notation What interval is positive What interval is negative How do you know

Find dy da in terms of x and y if cos 7y sin 7y y 14 0 da

Use the graph of y g x below to determine which of the choices could be the graph of y g x Hint Make a sign chart and interpret it first

Use the graph of y f x below to determine which of the choices could be the graph of y f x Hint Make a sign chart and interpret it first

A waste management company is designing a rectangular construction dumpster that will be twice as long as it is wide and must hold 20 yd of debris Find the dimensions of the dumpster that will minimize its surface area X

ROO OC

For y In 7 3x 9 to find y would require the chain rule If y f g x find an f x and g x that would allow you to use the chain rule

Determine where f x 6x5 40x 120 is increasing Select all that apply Hint Use a sign chart for the first derivative 2 2 0 2 0 0 2 DE CORRESA 19Aumar HNSTORERNA64 TERMINE ANCHEWIN

Find the derivative of the following 3 5 g x Og x 12a 5 5 Og x 3x 4 52 3 g x 04 g x 12 15 25 A 5 3

f x 0 if x 0 or 2 x 4 f x 0 if 0 x 2 orx 4 f x 0 if 1 x 3 f x 0 if x 1 orx 3 2

Suppose f is continuous on a If f 1 0 and f 1 1 what can you say about f At x 1 f has a local maximum At x 1 f has a local minimum At x 1 f has neither a maximum nor a minimum More information is needed to determine if f has a maximum or minimum at x 1 b If f 3 0 and f 3 0 what can you say about f At x 3 f has a local maximum At x 3 f has a local minimum At x 3 f has neither a maximum nor a minimum More information is needed to determine if f has a maximum or minimum at x 3

Let f t be the temperature at time t where you live and suppose that at time t 1 you feel uncomfortably hot What happens to the temperature in each case a f 1 2 f 1 2 The temperature is increasing and the rate of increase is increasing O The temperature is increasing but the rate of increase is decreasing O The temperature is decreasing but the rate of change is increasing becoming less negative The temperature is decreasing and the rate of change is decreasing becoming more negative b f 1 2 f 1 2 O The temperature is increasing and the rate of increase is increasing The temperature is increasing but the rate of increase is decreasing The temperature is decreasing but the rate of change is increasing becoming less negative The temperature is decreasing and the rate of change is decreasing becoming more negative c d f 1 2 f 1 2 The temperature is increasing and the rate of increase is increasing The temperature is increasing but the rate of increase is decreasing The temperature is decreasing but the rate of change is increasing becoming less negative The temperature is decreasing and the rate of change is decreasing becoming more negative f 1 2 F 1 2 The temperature is increasing and the rate of increase is increasing MY NO The temperature is increasing but the rate of increase is decreasing The temperature is decreasing but the rate of change is increasing becoming less negative The temperature is decreasing and the rate of change is decreasing becoming more negative

6 Find every number c that satisfies the conclusion of the Mean Value Theorem for the function f x x 4x x on the interval 0 1

4 1 Maximum and Minimum Values 1 For the following functions find all critical numbers exactly a f x x5 2x T b f x x 2 sin x for 0 x 2

Find u v U V u i 4j v 6i j

1 For the function F x 5 asymptotes 1970 a graph the rational function using transformations b use the final graph to find the domain and range and c use the final graph to list any vertical horizontal or oblique X B There is one vertical asymptote It is x 0 Type an equation Use integers or fractions for any numbers in the equation C There is no vertical asymptote What is are the horizontal asymptote s Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The top horizontal asymptote is The bottom horizontal asymptote is Type equations Use integers or fractions for any numbers in the equations B There is one horizontal asymptote It is y 5 Type an equation Use integers or fractions for any numbers in the equation OC There is no horizontal asymptote What is are the oblique asymptote s Select the correct choice below and if necessary fill in the answer box es to complete your choice 0 67 of 1 O A There is one oblique asymptote It is Type an equation Use integers or fractions for any numbers in the equation OB The oblique asymptote with the positive slope is and the oblique asymptote with the negative slope is Type equations Use integers or fractions for any numbers in the equations C There is no oblique asymptote Sa

A random variable X is uniformly distributed between a 3 and b 13 as shown below 2 a 3 4 5 6 7 8 9 10 11 12 13 14 Find the height h of the rectangle that corresponds to the sample space 1 Recall that if X is uniformly distributed from a to b then h b a

1 For the function F x 5 a graph the rational function using transformations b use the final graph to find the domain and range and c use the final graph to list any vertical horizontal or oblique X asymptotes A The domain of the given function is xlx is a real number x Type an integer or a simplified fraction B The domain of the given function is xlx is a real number x 0 Type an integer or a simplified fraction Use a comma to separate answers as needed C The domain of the given function is xlx is a real number x Type an integer or a simplified fraction D The domain of the given function is the set of all real numbers What is the range of the given function Select the correct choice below and if necessary fill in the answer box to complete your choice OA The range of the given function is yly is a real number y Type an integer or a simplified fraction OB The range of the given function is yly is a real number y Type an integer or a simplified fraction Use a comma to separate answers as needed OC The range of the given function is yly is a real number y Type an integer or a simplified fraction D The range of the given function is the set of all real numbers

Solve the given equation Enter your answers as a comma separated list Let k be any integer Round terms to two decimal places where appropriate cos 8 rad