# Differentiation Questions and Answers

Calculus

DifferentiationOil leaked from a tank at a rate of r t liters per hour The rate decreased as time passed and values of the rate at two hour time intervals are shown in the table In this problem we are going to estimate the total amount of oil that leaked out during this ten hour time period t hours 0 2 4 6 8 10 r t L hr 9 4 7 7 7 3 5 9 4 8 4 1 Rate of Oil Leakage Left Endpoint Approximation Estimate the total amount of oil that leaked out using left endpoints Total Amount of Oil Right Endpoint Approximation Estimate the total amount of oil that leaked out using right endpoints Total Amount of Oil liters liters

Calculus

DifferentiationThe limit below is a definition of f a Determine the function f x and the value of a lim h 0 f x 2 5 h 0 4 h

Calculus

DifferentiationDetermine whether lim f x exists If so give the limit X 2 What is the value of the limit Select the correct choice below and if necessary fill in the answer box to complete your choice A The limit is OB The limit does not exist 3 2

Calculus

DifferentiationPerform a first derivative test on the function f x x 5x 2 5 5 a Locate the critical points of the given function b Use the first derivative test to locate the local maximum and minimum values c Identify the absolute maximum and minimum values of the function on the given interval when they exist Simplify your answer OB There is no local minimum 2 Locate the maximum value Select the correct choice below and if necessary fill in the answer box within your choice OA There is a local maximum at x Simplify your answer B There is no local maximum A c Identify the absolute minimum value of the function Select the correct choice below and if necessary fill in the answer boxes to complete your ch The absolute minimum value is Simplify your answer OB There is no absolute minimum 33 4 Identify the absolute maximum value of the function Select the correct choice below and if necessary fill in the answer box within your choice OA The absolute maximum value is Simplify your answer OB There is no absolute maximum

Calculus

DifferentiationDirections Use the equation given below to answer the following question Derivative Use implicit differentiation to calculate da dy Note that we are calculating the derivative of z with respect to y the opposite of what we normally dy This means we need to regard y as the independent variable and a as the dependent variable x y xy 2xy 0 da

Calculus

DifferentiationDirections Use the equ Part 1 Derivative dy Calculate the derivative by implicit differentiation da dy da Part 2 Evaluation Evaluate the derivative at the point 0 2 dy da 0 2 x 4xy y 4

Calculus

DifferentiationDetermine the point on the graph of the unit circle that I I 2 corresponds to Then findcos andsin and state 2 which functions are undefined for I cos 2 KIN 0 2 I 2 The point on the unit circle is 0 1 Type an ordered pair Type an integer or a simplified fractio T T Simplify your answer sin 1 Simpl 2 1 0 2 cotangent tangent Ay 0 1 0 0 1 1 0 O

Calculus

Differentiationotated through 180 Assume the radius of the bicycle wheel is 12 6 inches icycle will travel approximately in nd to the nearest tenth 1 45 in

Calculus

DifferentiationCompute the difference quotient f x h f x h f x h f x h for the function f x 3x 2x 4 Simplify your answer as much as possible

Calculus

DifferentiationFind the derivative of f x f 9 1 at x X II 7 l os 9 Type an integer or a simplified fraction

Calculus

DifferentiationUse the formula s rot to find the value of the missing variable T 4 radians per sec r 12 cm t 16 sec S cm Type an exact answer using as needed

Calculus

DifferentiationThe tangent line to the graph of y x at the point x y has slope 3x Using this result find the slope of the curve at the point 3 27 The slope of the curve at the point 3 27 is

Calculus

DifferentiationConvert the following angle to degrees minutes and seconds form a 83 7022 The answer is 0 Simplify your answers Round to the nearest second as needed

Calculus

DifferentiationWrite the equation of the tangent line to the graph of y x at the point x 0 9 Tyne an equation

Calculus

DifferentiationDirections Consider the following function f x at a 8 Part 1 Difference Quotient If h 0 then the difference quotient can be simplified into the form f 8 h f 8 8 h where A and B are constants Find these constants A and B Note It s possible for one or more of these constants to be 0 f x x 7 f 9 1 VA h B Part 2 Derivative Use the simplified expression from Part 1 to then calculate the derivative f 8 f 8 lim h 0 That is 1 A h B f 8 h f 8 h

Calculus

DifferentiationDirections Use the graph of function f x given below to estimate the following values of its derivative J x Note that a portion of the tangent line has been drawn for you at x 2 1 Estimate the value of f 4 2 Estimate the value of f 2 3 Estimate the value of f 1 ello 5 4 f x 2 3 4 5 H

Calculus

DifferentiationPart 1 Difference Quotient If h 0 then the difference quotient can be simplified into the form That is f x f x h f x h Note It s possible for one or more of these coefficients to be 0 1 x 4 1 x 4 A Bh 1 x 4 A Bh where the coefficients A and B can be functions of a or just constant terms Find these coefficients A B Part 2 Derivative Use the simplified expression from Part 1 to then calculate the derivative f x f x lim h 0 f x h f x h

Calculus

DifferentiationCompute the following limits Algebraically 1 1 2 x x 2 X 2 a lim 4 x h 4 x 2 c lim wallot 2 0 0 0csc0 b lim bir 1019 digs brin alogad

Calculus

Differentiation4 Each angle measure is given in radians Write the corresponding angle measure in degrees a 0 18 4 3 d 0 8 b 0 C 0 716

Calculus

DifferentiationFind the length and direction when defined of u xv and vxu u 21 2j 3k v 3i 3k The length of uxvis 9 Type an exact answer using radicals as needed Select the correct choice below and if necessary fill in the answer boxes to complete your choice 1 3 Type exact answers using radicals as needed The direction of u xv is 2 3 i OB The direction of u xv is undefined j 2 3 The length of vx u is Type an exact answer using radicals as needed k

Calculus

DifferentiationFind the length and direction when defined of u xv and vxu u 2i 2j 3k v 3i 3k The length of u x v is Type an exact answer using radicals as needed

Calculus

Differentiation6 An angle in standard position has a measure of 5 8 radians In which quadrant is the terminal side of the angle

Calculus

DifferentiationConsider the velocity function v t 4 sint for st 2 Complete the sentence The same distance could have been traveled over the given time period at a constant velocity of The same distance could have been traveled over the given time period at a constant velocity of Type an exact answer in terms of

Calculus

DifferentiationGiven that g x a g 9 X 8 x 3 b g 8 find each of the following c g 3 d g 15 25 Simplify your answer a Select the correct choice below and if necessary fill in the answer box to complete your choice e g x h OA 9 9 OB 9 9 does not exist OC g 9 is not a real number b Select the correct choice below and if necessary fill in the answer box to complete your choice OA 9 8 Simplify your answer OB g 8 does not exist OC 9 8 is not a real number c Select the correct choice below and if necessary fill in the answer box to complete your choice OA 9 3 Simplify your answer OB 9 3 does not exist Oc 9 3 is not a real number

Calculus

DifferentiationVerify that the trigonometric equation is an identity cos a sec a cos acsc a csc a Which of the following statements establishes the identity OA cos a sec a cos2acsc a cos x OB OC cos a sec a cos x csc cos x 2 cos2x sec cos2a csc sin x 1 COS 1 2 COS 1 2 COS 1 sin 2 sin 1 2 sin cos 2 cos COS 2 sin x 2 1 2 a sin 1 2a sin 1 2 2 cos a sin csc 2 csc csc x

Calculus

Differentiation12 13 15 Write each expression in radical form 16 7 12 56 A b C 5b Simplify 13 64x4 A x4 C 512x6 14 14 Tm Write each expression in exponential form 1 A 3m C 7m Simplify 3 15 x 10 5 A x5 C 10x B 106 D 2b B 1024x 5 D 6x 1 16 6b A 6b C 376 B m D m B 27x D x Write each expression in radical form B 46 D 3b 2024 Kuta Softw a re LLC All rights reserved Made

Calculus

DifferentiationFind the area of the region R bounded by the graph of f x 9 x and the x axis on the interval 9 18 Graph and show the region R Choose the correct graph below The region R is shown in yellow O A 36 36 Ay 36 X 36 OB Ay 36 36 36 Q O C 36 36 Ay 36 36 OD 36 in

Calculus

Differentiation2 sin 2t Using a table of values the limiting value is to sin 2t 2 t cos 2t Consider lim Submi Enter DNE if the limit does not exist

Calculus

DifferentiationTutorial Exercise Determine the infinite limit lim X 10 x 9 X 9 Step 1 As x approaches 9 from the left the numerator approaches Also as x approaches 9 from the left the denominator approaches from the Select

Calculus

DifferentiationLet a and b be positive numbers such that the function ax x 2 0 x 2 f x bx a Then write the value of f 8 3 x 2 is differentiable for all x 0

Calculus

DifferentiationIdentify the transformations of the graph of f x that produce the graph of the given function g x Then graph g x on the same coordinate plane as the graph of f x by applying the transformations on the asymptotes x 0 and y 0 to the reference points 1 1 and 1 1 Also state the domain and range of g x using inequalities g x 2 1 2 6 5 4 3 4 5

Calculus

DifferentiationGraph the equation using the slope and the y intercept 10 y 3x 2 Use the graphing tool to graph the line Use the slope and y intercept when drawing the line Click to enlarge graph 20 16 12 8 20 16 12 2 ob 8 4 4 8 12 8 12 16 X 20 Q

Calculus

DifferentiationUse the graph of f x log 3x to graph the function f x log 3 x 2 4 Use the graphing tool to graph the function Click to enlarge graph

Calculus

DifferentiationFind the derivative f ex for each function given be a f x 1 e 1 ex x b f x Inx fex 5x3 Cosx sinx 4

Calculus

DifferentiationGiven g x is a transformation of the graph f x 2x determine which of the below correctly describe g x f x 2 54 3 2 1 N 7 21 3 S Asymptote y 0 123 g x 2 x g x 2 x g x 2 x g x 2x 1 Asymptote y 0 5 4 3 2 1 21 3 g x 1 2 3 4

Calculus

Differentiation3 Match the graph of each function in a d with the graph of 3 Match the graph of each function in a d with the graph of its derivative in 1 IV Clive reasons for your choices A VA A c 111 0 VA d

Calculus

Differentiation98 Answer the questions below about the function whose derivative is f x 2 2 x 0 X a What are the critical points of f b On what open intervals is f increasing or decreasing c At what points if any does f assume local maximum and minimum values