Calculus Questions

Consider the marginal cost function C x 400 12x 0 03x a Find the additional cost incurred in dollars when production is increased from 150 units to 200 units b Find the additional cost incurred in dollars when production is increased from 400 units to 450 units

a Determine whether the Mean Value Theorem applies to the function f x 7 x on the interval 1 2 b If so find the point s that are guaranteed to exist by the Mean Value Theorem a Choose the correct answer below OA No because the function is continuous on the interval 1 2 but is not differentiable on the interval 1 2 OB No because the function is differentiable on the interval 1 2 but is not continuous on the interval 1 2 OC No because the function is not continuous on the interval 1 2 and is not differentiable on the interval 1 2 D Yes because the function is continuous on the interval 1 2 and differentiable on the interval 1 2 b Select the correct choice below and if necessary fill in the answer box to complete your choice OA The point s is are x www Simplify your answer Use a comma to separate answers as needed B The Mean Value Theorem does not apply in this case

Consider 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

Simplify Assume that all expressions under radicals represent nonnegative numbers 15a 6 10a 15a 6 79

der an object moving along a line with the following velocity and initial position v t 6 3t on 0 4 s 0 0 Determine the position function for t20 using both the antiderivative method and the Fundamental Theorem of Calculus Check for agreement between the two methods To determine the position function for t20 using the antiderivative method first determine how the velocity function and the position function are related Choose the correct answer below A The position function is the antiderivative of the velocity function OB The position function is the absolute value of the antiderivative of the velocity function OC The position function is the derivative of the velocity function OD The velocity function is the antiderivative of the absolute value of the position function Which equation below will correctly give the position function according to the Fundamental Theorem of Calculus A s t s 0 t Sv x dx 0 MIN b C s t s 0 Sv t dt b B s t v t dt t OD s 0 s t v x d Sv x dx Determine the position function for t20 using both methods Select the correct choice below and fill in the answer box es to complete your choice OA The same function is obtained using each method The position function is s t OB Different functions are obtained using each method The position function obtained using the antiderivative method is s t of Calculus is s t 0 and the position function obtained using the Fundamental Th

ven that g x X 8 x 3 b g 8 find each of the following c g 3 d g 15 25 e g x h

Given 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

Establish the identity 1 sin 0 sec 0 1 Which of the following shows the key steps in establishing the identity O A OB 1 sin 20 sec 0 csc 0 cos 0 1 sin 20 sec 0 tan 0 sec 0 1 sin 20 1 sin20 sec 0 cos 0sec 0 1 2 sec 0 cos 0 cos 0 cot 0 sec 0 tan 0 20 1 OC 1 sin20 sec 0 sec 0 sin 0 sec 0 sec 0 tan OD 1 sin e sec 0 cos 20 cos 20 1

onsider an object moving along a line with the following velocity and initial position v t 6 3t on 0 4 s 0 0 etermine the position function for t20 using both the antiderivative method and the Fundamental Theorem of Calculus Check for agreement between the two methods to determine the position function for t20 using the antiderivative method first determine how the velocity function and the position function are related Choose the correct answer below YA The position function is the antiderivative of the velocity function OB The position function is the absolute value of the antiderivative of the velocity function OC The position function is the derivative of the velocity function D The velocity function is the antiderivative of the absolute value of the position function Which equation below will correctly give the position function according to the Fundamental Theorem of Calculus t A s t s 0 v x dx 0 b OC s t s 0 v t b OB s t v t dt Sv t d a Sv x dx 0 OD s 0 s t 0

Verify 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

Consider an object moving along a line with the following velocity and initial position v t 6 3t on 0 4 s 0 0 Determine the position function for t20 using both the antiderivative method and the Fundamental Theorem of Calculus Check for agreement between the two methods CHI To determine the position function for t20 using the antiderivative method first determine how the velocity function and the position function are related Choose the correct answer below OA The position function is the antiderivative of the velocity function OB The position function is the absolute value of the antiderivative of the velocity function OC The position function is the derivative of the velocity function D The velocity function is the antiderivative of the absolute value of the position function

Simplify the difference quotient f x x 15x f x f a x a f x f a x a Simplify your answer for the following function

Use the table to evaluate the given compositions a h g 1 b g f 4 d g h f 1 e f f f 1 h g f h 1 g f h g 0 j f f h 1 c h h 2 f h h h 4 i g g g 3 a h g 1 b g f 4 c h h 2 d g h f 1 Simplify your answer e f f f 1 Simplify your answer f h h h 4 g f h g 0 h g f h 1 Simplify your answer Simplify your answer Simplify your answer Simplify your answer Simplify your answer Simplify your answer 1 g g 9 3 Simplify your answer j f f h 1 Simplify your answer X f x g x h x 1 3 0 3 4 0 1 2 1 1 2 a 4 3 0 2 1 0

If cos 0 3 5 and 0 0 then tan 0 O 3 4 O 4 5 O 4 3 O 5 4

For what value of t does 2 t 2 3 5 3 t 1 O There is no value of t that works O 1 5 05 O 1 5 0 5 O 3 5

For what values of x does 0 none of these 0z 1 orz 2 Ox 1 0 1 x 2 0x 2

If log x 5 3 then a 08 O none of these O 13 0 1032 5 O log2

Solve the equation for a 0 1 4 Othere are no real solutions 0 2 0 2 2 2 0 2 4 2 x 4x s

An equation of the line passing through the points 1 3 and 5 2 is given by Oy 19 x 10 6 O none of these Oy x Oy fx 3 28 6 O y 5 fx 1 13 il 6

Choose the function g x whose graph is obtained by transforming the the graph of f x as follows the graph of f x x shifted 3 units to the right and 2 units up O g x x 2 3 O g x x 3 2 O g x x 2 3 O none of these O g x x 3 2

What is the radian measure for an angle whose degree measure is 240 O 47 3 6 3 OT 3 0 277 3

If f x 2x 5 and g x 1 a then g f 2 0 3 0 80 01 0 1 0 3 O 80

Simplify the expression of 0 30 1 01 3 x h 4 3x 4 h

12 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

Consider an object moving along a line with the given velocity v Assume t is time measured in seconds and velocities have units a Determine when the motion is in the positive direction and when it is in the negative direction b Find the displacement over the given interval c Find the distance traveled over the given interval v t 212 16t 30 0 6

At noon t 0 Alicia starts walking along a long straight road at 5 mi hr Her velocity decreases according to the function v t 5 t 1 for t 0 At noon Boris also starts walking along the same road with a 1 mi head start on Alicia his velocity is given by u t 3 t 1 for t20 a Find the position functions for Alicia and Boris where s 0 corresponds to Alicia s starting point b When if ever does Alicia overtake Boris

The figure shows the velocity function for motion along a line Assume the motion begins with an initial position of s 0 0 Determine the following a The displacement between t 0 and t 10 b The distance traveled between t 0 and t 10 c The position at t 10 4 2 0 2 t G

Consider the graph shown in the figure which gives the velocity of an object moving along a line Assume time is measured in hours and distance is measured in miles The areas of three regions bounded by the velocity curve and the t axis are also given Complete parts a through e a On what intervals is the object moving in the positive direction Select the correct choice and if necessary fill in the answer box to complete your choice OA The object is moving in the positive direction on the interval s Type your answer in interval notation Use a comma to separate answers as needed The object is never moving in the positive direction OB M 6 A 15 11 15

Find 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

Choose the correct graph below The region R is shown in yellow A 18 48 O Points 0 a the region R bounded by the graph of f x 12 x and the x axis on the interval 12 24 Graph and show the region R 48 The area of R is B 48 Ay 48 O C 78 Ar P

the result of integrating a population growth rate between times t a and t b where b a Choose the correct answer OA Integrating a population growth rate results in the net change in the population growth rate from time a to time B Integrating a population growth rate results in the rate of change of the population from time a to time b OC Integrating a population growth rate results in the net change of the population from time a to time b

3 Compute the following limits h 3 h 9 h 9 b lim b lim cos x sin x X R

4 x Let f x x ifx 2 if 2 x 2 a 6 x ifx22 Sketch the graph of f 2 2 10 2 00 Use the graph to determine the values of a for which lim f x does not exist Enter your answers as a comma separated list O 2 3 2 O

2 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

The figure below shows a fixed circle C with equation x 1 2 y2 1 and another shrinking circle C centered at the origin with positive y intercept P 0 r Let Q be the point of intersection between the two circles pictured draw a line through P am Q and let R be the x intercept of that line b The line through P and Q has equation y a Find the coordinates of the point Q your answers will involve r Q c The point R lim R d 20 y X O C R

A circle of radius r centered at the point 0 r in the plane will intersect the y axis at the origin and the point A 0 2r as pictured below A line passes through the point A and the point C 3 2 0 on the x axis In this problem we will investigate the coordinates of the intersection point B between the circle and the line as roo A 0 2r Y 0 0 B a The line through A and C has equation b The x coordinate of the point B is c The y coordinate of the point B is d The limit as roo of the x coordinate of B is e The limit as roo of the y coordinate of B is if your answer is oo write infinity if your answer is oo write infinity

Tutorial 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

Use the given graph off to state the value of each quantity if it exists If an answer does not exist enter DNE a lim f x X 3 b 2 lim f x x 3 c lim f x d lim f x X 7 0 0 e f 7 y 6 N 2 4 6 8

Use the given graph off to state the value of each quantity if it exists If an answer does not exist enter DNE a lim f x x 2 b lim f x c lim f x x 2 d f 2 e lim f x X 4 f f 4 2 0 2 4

Let cos x a Identify a Choose the correct graph below OB O A O D d O C x

The graphs of the functions f and g in the figure are obtained by vertical and horizontal shifts and scalings of y x Find formulas for f and g Verify your answers with a graphing utility f x Type an expression using x as the variable y xl x g x 8 f x X

2 log b log b xy xy N Z Type an integer or a decimal www

First rewrite the expression log c using base 10 log c

Expre 54 Do not evaluate

are associated with the points P and Q on the graph of the function Answer parts a and b 10 The volume V of a gas in cubic centimeters is given by V P atmospheres and 0 5 sps4 a Sketch a graph of the function and the secant line through P and Q b Find the slope of the secant line in part a and interpret your answer in terms of an average rate of change over the interval Include units in your answer a Choose the correct graph below A AV 20 16 12 8 00 4 0 0 1 2 3 4 5 6 p B AV 20 16 12 8 4 P 0 0 1 2 3 4 5 6 b The slope of the secant line is rate of Type integers or decimals OC 20 16 6284 AV 12 where p is the pressure in This means the over the interval 0 5 p 4 p Q 0 1 2 3 4 5 6 G D AV 20 16 12 8 4 OFFI 0 1 2 3 4 5 6 changes at an average

Let A x be the area of the region bounded by the t axis and the graph of y f t from t 0 to t x Consider the given function and graph f t 8 a Find A 8 b Find A 16 c Find a formula for A x a A 8 Simplify your answer Ay 3 2 1 0 16

28 Do not evaluate

In a study of anxiety a group of participants shows a mean score of 35 with a standard deviation of 5 What is the percentage if the score is between 36 and 42 A 11 998 B 22 998 C 33 998 D 44 998

a Make a table of the factorial function for n 1 2 3 4 5 b Graph these data points and then connect them with a smooth curve c What is the least value of n for which n 106 a Complete the table below 1 n f n n OA 140 b Choose the correct graph below Af n 0 1 0 6 2 2 n OB Af n 140 3 6 0 4 24 6 n O C Af n 140 5 120 0 0 6 OD Af n 140 0 0 6 n

A function and an interval of its independent variable are given The endpoints of the interval are associated with the points P and Q on the graph of the function Answer parts a and b 10 The volume V of a gas in cubic centimeters is given by V where p is the pressure in atmospheres and P 0 5 sps4 a Sketch a graph of the function and the secant line through P and Q b Find the slope of the secant line in part a and interpret your answer in terms of an average rate of change over the interval Include units in your answer a Choose the correct graph below O A 20 16 12 628 4 0 0 1 2 3 4 5 6 O B Q 20 Q16 12 8 4 ABE 0 0 1 2 3 4 5 6 Q O C 20 Q16 12 8 4 C 0 0 1 2 3 4 5 6 O D Q AV 20 Q16 12 8 N 4 0 0 1 2 3 4 5 6 P Q