Calculus Questions

The graph of y f x solid and y g x dashed is shown 6 5 4 3 4 lim 2 1 9 x 0 8 f x lim 24 1 g z Does Not Exist 88 0 8 2 z 1 g z Does Not Exist 0 Find the following limits 1 Does Not Exist 6 5 4 3 2 1 2 3 nd 5 6 2 3 4 1 1 1 1 S 6 q

Let f t be the temperature of a cup of coffee t minutes after it has been poured Interpret f 8 110 and f 8 5 Estimate the temperature of the coffee after 8 minutes and 42 seconds that is after 8 7 minutes C What does f 8 110 imply OA 8 minutes after the coffee has been poured the temperature of the cup of coffee is rising at a rate of 110 degrees per minute B 110 minutes after the coffee has been poured the temperature of the cup of coffee is 8 degrees O C 110 minutes after the coffee has been poured the temperature of the cup of coffee is rising at a rate of 8 degrees per minute OD 8 minutes after the coffee has been poured the temperature of the cup of coffee is 110 degrees

An analysis of the daily output of a factory assembly line shows that about 70t t At t 6 the rate of production is units per hour 1 t3 units are produced after t hours of work 0 st 8 What is the rate of production in units per hour when t 62 108 CL

Suppose that f x x a What is the average rate of change of f x over each of the following intervals 3 to 4 3 to 3 5 3 and to 3 1 b What is the instantaneous rate of change of f x when x 3 a The average rate of change of f x over the interval 3 to 4 is Simplify your answer

Theo and Sasha start at the same place on a straight road riding bikes with the following velocities measured in mi hr Complete parts a through f VT 1 5 for 120 Vg t 10t for Osts2 and vg t 20 for 1 2 Theo Sasha Graph the velocity functions for both riders Choose the correct graph below JA 25 Av Q C OB Av 25 04 OC O D 25 Av

Find the following lim cos 7a sin 0 0 cos 72 cos sin 72 sin cos 72 sin 7x sin cos 7x cos cos 72 sin sin 72 cos cos 72 sin sin 72 COS Consequently cos 72 sin 72 cos 7x cos 71 b sin 7x 8 Part 2 Part 3 o Part 4 of

CURRENT OBJECTIVE Use the washer method to find the volume of a solid of revolution around the x axis Question Let R represent the region bounded above by the graph of f x 4x 1 below by the graph of g x 2 and on each side by x 1 and x 4 What is the volume of the solid of revolution obtained by rotating R about the x axis Enter an exact value in terms of dow

6 Here is a graph of a pattern of numbers where V is a function of n The first point is 0 1 22 20 18 16 14 12 10 8 6 28 26 24 4 2 32 30 V 2 3 4 IMMathematics 5 a Plot the next 2 points on the graph that follow the pattern b Write an equation to describe the relationship between V and n f 1 1 f n f n 1 3 n 41 3 8

Question Let the curve C be the function f y 6y5 1 over the interval 1 y 7 Set up the integral for the arc length of the curve C Select the correct answer below 900y 1 dy T 1 30y dy 1 6y 2 dy 6y 1 1 dy of oy 1 dy GOLA 1 du

The velocity of an airplane flying into a headwind is given by v t 30 25 12 mi hr for Osts3 hr Assume that s 0 0 a Determine the position function for 0 t 3 b How far does the airplane travel in the first 2 hr c How far has the airplane traveled at the instant its velocity reaches 500 mi hr

The graph below is the function f x 8 7 6 5 4 3 2 8 7 6 S 4 lim 118 32 4 12 2 3 3 4 5 6 7 4 6 8 1 2 3 4 5 6 7 8 96 f x ter the limit If the limit is m enter go If the limit is enter oo If no limit exists

A cyclist rides down a long straight road at a velocity in m min given by v t 100 10t for 0 st 10 a How far does the cyclist travel in the first 4 min b How far does the cyclist travel in the first 8 min c How far has the cyclist traveled when his velocity is 75 m min

Given lim f x 2 and lim g x 1 evaluate 8118 X118 lim g x ln 6f x 4 60 8118 lim 69 x e f z 30 8118 lim 8118 lim 8118 6f x 49 a cos f x g x 11 2 1800 20 X X X X

Question Consider the curve f y y for 1 y 4 Find the arc length of the curv

Given lim f x 7 and lim g x 4 evaluate 8118 lim 2f x 4g x 8118 lim 2 4f x 8118 lim x418 2f x 4g x 2 4f x 2 lim 2f x 4g x 2 4f x 8118 11

Question Find the volume of a solid of revolution formed by revolving the region bounded by the graphs of f x xand g x 1 over the interval 1 3 around the x axis Hint graph the functions to determine which graph forms the upper bound and which graph forms the lower bound then use the washer method Provide your answer below

Find the following values for f x sin Be sure your computing device is set to RADIAN mode X f x 0 8105 Based on these values O lim sin x 0 lim sin x 0 Olim sin x 0 0 0123 70 X 05 does not exist 0 X 0 5723 0 049453719922586 0 00191 X 0 6136 0 000127 X

Check all that apply sin 5x sin CT 5 5x 5x sin 6x 5x 6x sin 5x sin 6x 6 sin 5x 6x

Question 1 Consider the curve f x x for 1 x 2 Find the arc length of the curve 2x Provide your answer below

Suppose 0 If u 2x then lim u x 0 lim 1 0 lim I 0 lim A 0 sin u u lim 0 sin u 1 u sin 2x x sin 2x 3x

lim sin x 4 sin 4 T

The graph of y f x solid and y g x dashed is shown AT 4 3 Find the following limits f x lim 2 4 g x 0 lim f x lim 2 4 g z 8 Does Not Exist 0 2 1 f x g z Does Not Exist 6 5 4 3 2 1 1 2 3 A 5 6 1 2 3 4 5 6

State whether the function is continuous at the given x values Justify your answers sqvt done lar 2x 1 x 1 7 f x 2 2 1 x 4 x 4 a x 3x at x 1 b at x 4 2 4 1 4 3 4 3 4

The graph of y f x solid and y g x dashed is shown 6 5 4 B Find the following limits f x lim 1 g x 818 0 Does Not Exist f x lim 2 1 g x 00 Does Not Exist 1 f z lim 2 1 g x 6 5 4 3 2 1 2 3 4 5 6 3 4 5 6

Find the limit sin 7x lim x0 sin 6x lim x 0 tan 6x sin 7x

To find lim 2 0 cos x 1 which of the following steps might be useful 8 Multiplying numerator and denominator by Adding 1 to numerator and denominator Multiplying numerator and denominator by cos x 1

Evaluate 3 2 3x lim 4x sin 2 0

Find the following lim cos 72 7x x 0 Question Help Written Example

sin 2x Which of the following is equal to sin 3x 2x 3x 2x sin 2x 2x sin 3x 3x sin 2x 3x sin 3x 2 sin x 3 sin x assume0 Select all that apply

Which of the following graphs exhibit ALL of the listed properties lim f x 2 818 lim f t o 2 f 2 is undefined 54 3 2 1 54 3 2 1 54 3 2 1 4 3 2 4 S 4 3 2 1 2 4 S 5 4 3 2 1 1 2 3 5 3 2 P DE 1 1 2 34 5 2 a 3 4 2 B q 70

Which of the following graphs exhibit ALL of the listed proper lim f x 2 HIX lim f x 00 I 1 f 1 2 54 3 2 1 51 4 13 2 1 1 2 3 4 54 5 54 3 2 1 4 3 2 1 2 3 4 st 3 2 54 3 2 1 2 1 2 3 1 2 3 4 5 La 4 5 1 2 3 4 a

The graph below is the function f x 8 7 16 15 lim f x 8118 2 1 8 7 6 5 4 3 2 1 2 3 34 4 5 56400 6 7 8 1 2 3 4 5 6 7 8

The graphs of y f x solid and y g x dashed are shown The two curves intersect at 5 0 Determine the sign of the following limits You may assume the limits exist lim f x g x is 5 Positive Negative Zero lim f x g x is 20 Positive Negative Zoro

Find the value s of k such that f x is continuous everywhere Sx 76 a 2 4x3 3kx 2 x 2 q f x

The graph of y f x has the following features f 1 0 lim f x 2 x 1 lim f x 1 1 Which of the following might be a graph of y f x a Q a

12 If the function f is continuous for all real numbers and if when x 3 then f 3 f x x 6x 9 x 3

x 3x 7 x 3 3 x 1 x 1 8 g x 2x 1 In x a at x 3 ran b at x

Suppose a vat contains 15 liters of water and 12 kg of salt At t 0 water is added at a rate of 20 liters minute while salt is added at a rate of 9 kg minute A function giving the concentration after t minutes will be C t express as a function of t As time passes the concentration will Decrease Increase to kg liter kg liter

Find the limit sin 5x 3x lim x 0 3x 40 sin 5x lim

The graph below is the function f x 8 7 6 5 4 3 2 lim f x 2 8 8 7 6 5 4 3 27 1 2 AWN 3 4 567 7 8 4 5 6 7 Enter the limit If the limit is o enter oo If the limit is enter oo If no limit exists enter

Suppose f x has the following values I f x lim 8 7 9 7 99 6 5 6 05 1 6 f T 7 999 8 6 005 9 appears to be 8 001 6 004 8 01 6 04 8 6

Given lim f x 2 and lim g x 1 evaluate 8118 8178 lim g x ln 6f x 4 8118 lim 8118 lim 8118 lim 8118 69 x e f z 6 2 49 x cos f x a x C

Suppose a student finds the following values of x and f x x If f 4 9 3 4 99 877 C 4 999 2393 these values would SUPPORT the claim that lim f x x 5

Evaluate the limit 4 6x lim I 11 4x Use oo if the limit is and oo if the limit is

Suppose lim f x 8 Find the following Type DNE if the limit does not exist 48 2 lim 8 f x 8 2 lim 2 8 f x 8 2 lim z 8 f x 8

Suppose a student finds the following values of x and f x X 2 9 f x 5 If f 2 99 971 2 999 8884 these values would make us DOUBT the claim that lim f x

B x y 11 Determine the exact values of cos I 4 TC 4 T 9 Use the diagram above to approximate the values of cosand sin 4 A 1 0 10 Use a calculator to find the values of cos and sin 4 4 and sin 4 12 Compare your answers to 9 11 What would account for any differences

Suppose lim f x 7 lim g x 3 x x x f x 5x 3 3x 1 lim 4 xxxg x

Curriculum 2 0 Precalculus Unit 3 Topic 1 SLT Evaluating Trigonometric Functions B x y TC G 6 7 Determine the exact values of cos A 1 0 5 Use the diagram above to approximate the values of cos TC 6 Use a calculator to find the values of cos and sin Name Aman Date cosand sin

PopQuiz 6 Prof Rudy Meangru Find the first derivative and do not simplify the Algebra y 1 e 1 e x