Differentiation Questions and Answers

7 A sink is filled with 54 quarts of water The function q f t 54 2 5t gives the volume of water in a sink i quarts t seconds after removing the stopper a Find f b What does the input of f represent What does the output of f represent c How long will it take for the sink to drain completely d How long will it take for the sink to lose one third of its water

Mrs Frasier worked her 40 regular hours last week plus 4 overtime hours at the double time rate Her gross pay was 734 40 What was her hourly rate

Mrs Roper works 40 hours per week regularly at a rate of 16 20 per hour When she works overtime her rate is time and a half of her regular hourly rate What is Mrs Roper s hourly overtime rate

x if x 1 f x 1if 1 x 2 x 1 ifx 2 5 a f 4 b f 1 c f 2 d f 5 1

Suppose that A Find an equation for the tangent line to the graph of fat x 2 Tangent line y f x 5x 2 B Find the average of all values of a where the tangent line is horizontal If there are no such values enter 1000 Average of a values

Find an equation of the tangent line to the curve at the point y Tangent line V y sin 5x cos 6x

Sketch the graph of f and use your sketch to find the absolute and local maximum and minimum values of f Enter your answer as a comma separated list If an answer does not exist enter DNE f x x 8 9x absolute maximum value absolute minimum value local maximum value s local minimum value s if 1 x 0 if 0 x 1

7 10 75 Points Verify the identity algebraically Use a graphing utility to check your result graphically 6 cot a 6 csc a 6 csc a 1 cot a DETAILS 6 cot a csc 1 Multiply the numerator and denominator by a common expression and then use a Pythagorean Identity to simplify Simplify your answers completely csc a 1 Need Help Submit Answer LARPRECALCRMRP7 5 2 058 6 cot a csc a 1 6 cot a csc a 1 Read It cot a 8 1 Points DETAILS Use the cofunction identities to evaluate the expression without the aid of a calculator MY NO LARPRECALCRMRP7 5 2 069 MI MY NOTES

Determine whether the given function is one to one If it is 656 16 810 1 2 5 5 3 3 1 6 5 is the function f 2 5 5 3 3 1 6 5 one to one Yes No

Sketch the graph of f and use your sketch to find the absolute and local maximum and minimum values of f Enter your answers as a comma separated list If an answer does not exist enter DNE if 1 x 0 x if 0 x 1 f x 8 9x absolute maximum value absolute minimum value local maximum value s local minimum value s Submit Answer Need Help Read It 3 0 11 Points Find the critical numbers of the function Enter your answers as a comma separated list If an answer does not print DETAILS SCALCET9 4 XP 1 022 JE DUB

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of 0 where 0 0 i 2 16 x x 4 cos 0

J 1 Find the following using the table below X 1 2 3 4 f x 2 3 4 1 f x 1 4 2 3 g x 1 2 4 3 g x 3 4 1 2 h 1 if h x h 1 if h x f x g x f x g x h 1 if h x f g x Question Hel 6 question

Find the derivative y x y ecosh 8x

Use the chain rule to find the derivative of 8 7x 9x Type your answer without fractional or negative exponents Use sqrt x for x

Find dy da dy da 11 for y Question Help In 3x 10 6e5t Video Written Exar

4 Let f x x 1 and g x 3x 7 a Find f g 2 b Write an expression for f g x c Find g f 2 d Write an expression for g f x

A particle moves along the x axis in such a way that its position at time t is given x 3t4 16t 24t2 for 5 t 5 a Determine the velocity and acceleration of the particle at time t b At what values of t is the particle at rest c At what values of t does the particle change direction d What is the velocity when the acceleration is first zero

3 Let f x x 51 13x 61 What is the value of f 1 Be sure to show your work

A particle moves along the x axis in such a way that its position at time t is given 4 by x 3t4 16t 24t for 5 t 5 a Determine the velocity and acceleration of the particle at time t

106 Find the derivative of f x ln x5 6x

104 If h x 91 f x find f x

Use the values to evaluate if possible all six trigonometric functions If an answer is undefined enter UNDEFINED 26 26 cot p 5 sin p cos q tan p csc p sec 4

Write out the first few terms of the geometric series the inequality holds and the series converges What is the second term of the given series 8 What is a n 0 The general geometric series is a ar ar ar ar What is the first term of the given series n to find a and r and find the sum of the series Then express the inequality r 1 in terms of x and find the values of x for wh 5 What is r 0 If the series is convergent what is the series sum Substitute the expression in x for r in the inequality r 1 to obtain 1 Isolate x For what values of x does the series com 1 0 1

Suppose that Find f x and then evaluate f at x 1 0 f 1 3x 1 f x 2x 1

This problem tests calculating new functions from old ones From the table below calculate the quantities asked for 64 3 0 2 1 62 528385 64 1 21 4 480501 f x g x 524418 62 2 22 6 476782 f x 24704 60 0 28 8 23188 g x 24578 56 2 26 8 23066 f 0 g 0 5 If h x f g x calculate h 3 f g 0 f f 3 f g 0

Let F x f f x and G x F x You also know that f 6 8 f 8 2 f 8 14 and f 6 14 Find F 6 and G 6

Find the length of the polar curve given by r 7e0 40 on the interval 0 0 Answer Graph the curve on a calculator Turn in a sketch of the curve

Consider the function f x x 6x on the closed interval 1 3 Find the exact value of the slope of the secant line over the interval m By the Mean Value Theorem there exists c in 1 3 so that m f c Find all values of such c in 1 3 Enter exact values If there is more than one solution separate them by a comma C

Consider the function f x over the interval 2 2 e4x Does the extreme value theorem guarantee the existence of an absolute maximum and minimum for f on this interval Select the correct answer below Yes No

1a A snowball melts so that its surface area decreases at a rate of 0 5 cm min Find the rate at which the radius decreases when the radius is 4 cm SA 4nr 1b A water tank filtration plant is built in the shape of an inverted cone with height 5 2 m and diameter 5 m at the top Water is being pumped into the tank at a rate of 3 3 1 2 m min find the rate at which the water level is rising when there is 8 m of water V h Find the equations of the tangent lines to the curve y x x 2 that pass through 1c the point 3 2 Include a sketch to illustrate your solution

Find the derivative of the function g x jaomani g x ex x 4 smal questi Add Work I et 3 4x You can retry this q

C x 120 6x find the marginal average cost when 10 units are proc

Find the area of the region bounded by the polar curve r 6e on the interval T 0 2

V Find the sum of the x and y intercepts of any tangent line to the curve x y 10

Find y x y dy da dax by implicit differentiation Note unfortunately WeBWork isn t quite smart enough to understand that x y for this problem is the same as ex y After differentiating d simplify beyond solving for dy

Find by implicit differentiation dy 1 dx tan x y Y 1 x dy da Remark once again WeBWork isn t quite smart enough to understand all correct answers Carry through the differentiation and solve for in the simplest way possible

is a radioactive material that decays according to the function A t Age scientist has a sample of 800 grams of strontium 90 a What is the decay rate of strontium 90 b How much strontium 90 is left after 20 years c When will only 200 grams of strontium 90 be left d What is the half life of strontium 90 a The decay rate of strontium 90 is 2 44 Type an integer or a decimal Include the negative sign for the decay rate b Approximately 491 grams of strontium 90 is left after 20 years Do not round until the final answer Then round to the nearest whole number as needed c Only 200 grams of strontium 90 will be left in about 56 8 years Do not round until the final answer Then round to the nearest tenth as needed 0 0244t d The half life of strontium 90 is approximately years Do not round until the final answer Then round to the nearest tenth as needed 1 Points 0 5 of 1 15 points where Ao is the initial amount present and A is the amount present at time t in years Assum

x 2 1 000 3 or 2 3 3 00 3 2 QUESTION 18 Solve the problem A fossilized leaf contains 35 of its normal amount of carbon 14 How old is the fossil to the nearest year Use 5600 years as the half life of 28 672 3474 8466 33 664

Find the following derivatives a dx arcsin x b arctan e dx Use inverse trigonometric functions to solve the following trigonometric equations c If the angle is known to be in the interval and 3 sin 2x 2 then x d The negative angle closest to zero that satisfies cos 0 3 7 is

Compute the following derivative d cos EEE

Find the derivative of the function It may be to your advantage to simplify before differentiating f x cos arcsin 52 SE3

Compute the following derivatives a b d da d da r arcsin r sin r arccos r

Let f x cos arcsin a a First find the derivative by using Chain Rule f x b Next we will find the derivative in a different way Rewrite f x without trigonometric functions Hint Set up a right triangle with a side x When you have finished this step f x can be written as f x Then find the derivative of this equivalent form of f

dr Let A be the area of a circle with radius r If dt dA dt 2 find when r 2

Form a polynomial whose zeros and degree are given Zeros 0 5 4 degree 3 f x x x x 20 for a 1 Of x x3 x2 20x for a 1 Of x x3 x2 20x for a 1 f x x x x 20 for a 1

Determine the maximum number of turning points of f f x x 2 x 5 3 0 000

Which of the following would not be a representation of the differentiable function f where f 5 3 6 points The slope of y f x at x 5 is 3 The instantaneous rate of change at x 5 of f x is 3 lim h 0 f 3 h f 3 h 5 The slope of a line tangent to the graph of fat x 5 is 3 Which of the following would not be a representation of the differentiable function f where f 5 3 6 points O The slope of y fx at x 5 is 30 The instantaneous rate of change at x 5 of f x is 3 Of 3 hf 3 lim h 0 h 5 The slope of a line tangent to the graph of f at x 5 is 3 Previous Question Next Question

Determine r x if r x ln 3 X COS X 7x 6

x 2 103 Find the derivative of f x 5x 1

101 Find f x of f x g z dy da 102 Find if y 5log3 1