Differentiation Questions and Answers

2 Solve analytically Express your answer in interval notation 3x 4 1 8

The radius of a circular disk is given as 29 cm with a maximum error in measurement of 0 2 cm a Use differentials to estimate the maximum error in the calculated area of the disk Round your answer to two decimal places cm b What is the relative error Round your answer to four decimal places What is the percentage error Round your answer to two decimal places

a Find the differential dy y 3 x dy b Evaluate dy for the given values of x and dx x 1 dx 0 1 dy

Determine whether the statement is true or false d dx True False In 10 1 10

Determine whether the statement is true or false 10x X10X 1 d dx True False

Groves Grices find dy Idr f x x 052 find dy dx y f x cos x g Sinxt 91 19

Given an equation sin y y x use implicit differentiation to find dy

Find the cloivative of a x x 1 2x 1 Find the denuatiua y ffe e 9x 12x 12 3x

61158 s x SC 6 St 473 9 d 5 5 be dx 1 T x the desallative of Find ftt 65E 4 t t q A B S x 6x 5e

10 Given y f x CSc cot In 9x

Sketch the graph of the funciton Include two full periods Use a graphing utility to verify your result Choose the correct graph y csc J 10 20 10 20 UUU MENT 10 20

Question 1 15 points Find the general solution of the following differen equation on the indicated interval dy dt 2 cos t y tan t 0 t 2

y sec 281 y sec

287 y cot 1 4 x

Suppose you are standing on the surface z f x y at the point a b f a b Interpret the meaning of f a b and fy a b in terms of slopes or rates of change Choose the correct answer below O A f a b is the slope of the surface in the direction parallel to the y axis and f a b is the slope of the surface in the direction parallel to the x axis both taken at a b OB f a b is the slope of the surface in the direction parallel to the x axis and fy a b is the slope of the surface in the direction parallel to the y axis both taken at x y OC f a b is the slope of the surface in the direction parallel to the x axis and fy a b is the slope of the surface in the direction parallel to the y axis both taken at a b OD f a b is the slope of the surface in the direction parallel to the y axis and fy a b is the slope of the surface in the direction parallel to the x axis both taken at x y

The speed of sound in m s in dry air is approximated by the function v T 331 0 67 where T is the air temperature in degrees Celsius Evaluate v T and interpret its meaning

1 Find the derivative of a b a y ln x 4 2x 5 b y ln cscx cotx c Find the differential dy y csc x

4 The legs of an isosceles right triangle increase in length at a rate of 2 m s Determine each X X Z a At what rate is the area of the triangle changing when the legs are 2 m long 1 b At what rate is the area of the triangle changing when the hypotenuse is 1 m long c At what rot

Find the differential of each function a y x sin 4x dy 4x cos 4x 2x sin 4x dx b y In 2 1 dy t 2dt 2tt X

A cylindrical tank with radius 7 m is being filled with water at a rate of 4 m min How fast is the height of the water increasing in m min m min

1 If two resistors with resistances R and R are connected in parallel as shown in the figure then the total resistance R measured in ohms 22 is given by R R 22 s www R 150 1 R 1 R If R and R are increasing at rates of 0 3 s and 0 2 n s respectively how fast in s is R changing when R 90 and R 100 2 Round your answer to three decimal places

Find by implicit differentiation dy dx dy dx II tan 5x y x 4xy 10yx 100y x 25y x 1 x 200x 8y 5x 43 Need Help Submit Answer 1 1 Points Read It DETAILS Watch It X

If x 3t t and y 7 5t find an equation of the tangent in terms of x and y to the curve at t 1 Tangent line

If x 5t4 5 and y 5t t find the following derivatives as functions of t dy dx d y dx

Graph the polynomial function 3 2 f x x 3x 13x 15 Choose the correct graph of the function below The scale for each graph is 10 to 10 for the x axis where a tick represents 1 unit and 65 to 65 for the y axis where a tick represents 5 units O A B Q O C y Q 27 5 O D Z Q Q G

Find the interval s of numbers that are less than a distance of 7 away from 8 Enter solution as an interval or union of intervals In absolute value notation where a represents the numbers this is OA z 81 7 OB 2 7 8 help intervals OC x 7 8 OD x 8 7 OE None of the above

a The graph of a function y f x is horizontally compressed by a factor of 4 Find an equation for this compressed function in terms of the function f x For example y 10f 9x 8 7 help equations b The graph of the function y e 10 is horizontally compressed by a factor of 4 Find an equation for this compressed function help equations c The graph of a function y g z is horizontally expanded by a factor of 9 Find an equation for this expanded function in terms of the function g x For example y 10g 9z 8 7 help equations d The graph of the function y 2 2z is horizontally expanded by a factor of 9 Find an equation for this expanded function help equations

5 8 points Consider the function 2 f x 2 i f 2 ii lim f x x 2 x 2 os TX k where k is a fixed constant The following is a graph for the function f x when k 3x if x 2 if x 2 MAMA a 4 points Use the graph above to determine the following If a limit does not exist write DNE iii lim f x x 2 10 iv lim f x x 2 b 2 points Find a value of k for which f x is continuous everywhere

Question 9 A box with a square base and no top is to be made from a square piece of carboard by cutting 8 in squares from each corner and folding up the sides The box is to hold 11552 in How big a piece of cardboard is needed Your answer is in by Question Help Video Message instructor in

Question 4 Find b and c so that y 7x bx c has vertex 9 8 b C

Score 0 42 0 10 answered Question 2 Start with the graph of f x 2 Then write a function that results from the given transformation Reflect f x about the y axis g x Question Help VIDEO Post to forum

Find the equation of the tangent line to the curve y 6x cos x at the point 7 6 The equation of this tangent line can be written in the form y mx b Compute m and b m EEB

2 Some values of a differentiable function f x and its derivative f x are given in the table below X 0 1 2 3 4 28 f x 0 5 3 0 5 3 1 5 3 2 f x 1 2 4 r f x a 6 points Compute g 1 where g x f x f t 2t dt Simplify your final answer

Use the figures below to evaluate the indicated derivative or state that it does not exist If the derivative does not exist enter dne in the answer blank The graph to the left in black gives f x while the graph to the right gives g x which is constant for values of a greater than 40 200 200 40 40 0 40 10 0 168 40 f g x 20 If the derivative does not exist enter dne 40 0 40 40 0 g x 168

Find y if y e 1 z

Let f x 12 sin x 9 cos x

6 Let f x x2 16 and g x 2x 9 a What is the domain of f What is the domain of g b Find g x f x c What is the domain of 9 2 f x

6 15 pts The average price C of Oreo Cookies was 2 69 lb in 1990 and 3 00 lb in 2018 Assume that the exponential growth model applies a Find the exponential growth rate in 5 decimal places b Find the exponential function that models this situation c What will the price of 1 lb of Oreo Cookies be in 2025 d Find the rate of change of the price of 1 lb of Oreo Cookies in 2025 and explain what this rate means

QUESTION 5 points Find the derivative of y 2x 1 x 3 4

2 17 pts Use limit properties algebraic simplification tables of values and or graphs to find each limit if it exists If you are using a table of values or a graph make sure you include the table of values or the graph If the limit does not exist explain why and be descriptive as possible x 2x 24 x 6 x 12x 36 a lim x 2x 24 b lim x 3x 8x 16 c lim 4 r 3 x 3

3 40 pts Find the derivative of the following functions a f x 4x 2x 0 4x 2x b f x el In 2x 3x c y d y x 2 ln x e F x x e x 2e

Find the quotient and remainder using synthetic division 6x x x 1 quotient

A graphing calculator is recommended Determine the end behavior of P P x x y y x x 13x as x as x

A company has a price demand function given by p x 30e 0 000 dollars per item when z items are sold Determine how many items the company must sell in order to maximize revenue items NOTE Round to 3 decimal places if needed

The graph of f a is given below Use this graph to determine the intervals where f a is increasing or decreasing click on graph to enlarge a Interval s where f a is increasing b Interval s where f a is decreasing help intervals help intervals

Product Rule d dx x g x f x g x g x f x Quotient Rule d f x dx g x g x x f x g x g x 24 f x 1 xex X

2 Find the Taylor series generated by fat x a for f x 3x x 2x x 2 at a 1

Product Rule dx f x g x f x g x g x f x Quotient Rule d f x g x f x f x g x dx 19 y v 20 v v 2t

Using the First Derivative Test what are the local extrema for the function h x 3x 9x Select the correct answer below 4 O There is a local minimum at x O There are no local extrema There is a local maximum at x There is a local minimum at x There is a local maximum at 9x 72x 9 2 and a local minimum at x 4 2 and a local maximum at x 4 2 and a local maximum at 4

3 The first tem fourth tom the number is 2 of of an 23 The terms arithmetic 2 sum of the series 28 0 A series is 7 and the the series 689 Find 13