Calculus Questions

Calculate the following limits at infinity If necessary use inf and inf to denote o and o Use DNE if a limit does not exist 1 lim sin x dne 818 2 lim sin x dne 8118 3 lim tan x 818 4 lim tan x 118 TC 2 TC 2 Part 2 Horizontal Asymptotes List any horizontal asymptotes for the following functions If there are no asymptotes enter NONE If there is more than one asymptote give a comma separated list i e 1 2 1 The function sin x has horizontal asymptote s y none 2 The function tan x has horizontal asymptote s y TC 2

Directions Consider the following function f z at a 2 Part 1 Difference Quotient It h 0 then the difference quotient can be simplified into the form Ah B That is 2 h f 2 Ah B h where A and B are constants Find these constants A 0 and B2 Note It s possible for one or more of these constants to be 0 Part 2 Derivative Use the simplified expression from Part 1 to then calculate the derivative f 2 f 2 2 f x 2x 6 where A and B are constants Find these constants A 0 and B 2 Note It s possible for one or more of these constants to be 0 f 2 2 f 2 lim h f 2 h f 2 J 21 J 2 h Part 2 Derivative Use the simplified expression from Part 1 to then calculate the derivative f 2 f 2 lim 40 Part 3 Tangent Line Find the equation of the tangent line to the curve y f a at a 2 Tangent Line y Ah B f 2 h f 2 h

Calculate the following limits at infinity for the given function f x If necessary use inf and inf to denote and Use DNE if a limit does not exist 1 lim f x 0 2 lim f x 818 1 200 f x 1 e has horizontal asymptote s y 0 1 Part 2 Horizontal Asymptotes List any horizontal asymptotes for this function f x If there are no asymptotes enter NONE If there is more than one asymptote give a comma separated list i e 1 2 1 The function f x 2

Directions Consider the following function f a at a 2 Part 1 Difference Quotient If h0 then the difference quotient can be simplified into the form 2 A f 2 h where A B and C are all non negative constants Find these constants A 2 and B49 and C 14 Note It s possible for one or more of these constants to be 0 1 f x 2x 3 2 f 2 49 Part 2 Derivative Use the simplified expression from Part 1 to then calculate the derivative f 2 B Ch f 2 49 B Ch f 2 lim A A 2 and B 49 and C 14 Note It s possible for one or more of these constants to be 0 That is note the negative in front f 2 h f 2 h Part 2 Derivative Use the simplified expression from Part 1 to then calculate the derivative f 2 f 2 lim h0 Part 3 Tangent Line Find the equation of the tangent line to the curve y f a at a 2 Tangent Line y f 2 h f 2 h

Calculate the following limits at infinity for the given function f x If necessary use inf and inf to denote and Use DNE if a limit does not exist 1 lim f x f x 4x x 15 x1 5 3x Part 2 Horizontal Asymptotes List any horizontal asymptotes for this function f x If there are no asymptotes enter NONE If there is more than one asymptote give a comma separated list i e 1 2 4x x 15

nd the trigonometric function values for 6x using the graph of the unit circle elect the correct choice below and if necessary fill in the answer box to complete your choice A sin 6 Simplify your answer fined 1 0 2 0 1 10 0 1 1 0

Find the circular function values for the given number s State which functions are undefined for s Do not use calculator 17x 2 SE COLX Select the correct choice below and if necessary fill in the answer box to complete your choice 1 0 Ay 2 0 2 0 1 0

Determine the point on the graph of the unit circle that corresponds to 3x 3 findcos andsin 2 2 and state which functions are undefined for The point on the unit circle is pimplified fraction 3x 2 Then 3x 2 1 0 2 Ay 0 1 0 1 0 0 1 x o

Use a calculator to approximate the following sin 6 3812 cos 6 3812

Determine the point on the graph of the unit circle that corresponds to 2x Then findcos 2x andsin 2x and state which functions are undefined for 2 The point on the unit circle is Typo an ordered pair Type an integer or a simplified fraction G 1 0 R 2 Ay 0 1 0 0 1 1 0 A o

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 Using the values shown in the graph the displacement of the object over the interval 0 6 is 4 miles c What is the total distance traveled by the object over the interval 2 10 Using the values shown d What is the displacement of the object over the interval 0 10 Using the values shown in the graph the displacement of the object over the interval 0 10 is 6 miles e Describe the position of the object relative to its initial position after 10 hours Select the correct choice and if necessary fill in the answer box to complete your choice the graph the total distance traveled by the object is 25 miles A After 10 hours the object is OB After 10 hours the object is FOOD mi in the negative direction from its initial position mi in the positive direction from its initial position returned to its initial position 11 2 4 15 16 10 8

Consider trying to evaluate the integral A A E B A ln x 1 O C Aarctan r A D x 4x 3x x 2 x 2 OE A ln x If you asked a computer algebra system for the answer which of the following terms would you know is wrong if it showed up in the answer it gave you WITHOUT even solving for the coefficients dx

Find the indefinite integral x 362 x 49 dx 0 0 C

K Find the length to three significant digits of the arc intercepted by a central angle 0 in a circle of radius r 2x 7 r 14 1 cm 0 radians The length of the intercepted arc is approximately cm Round to one decimal place as needed

otated 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

onvert the following degree measure to radians 120 120 radians Simplify your answer Type an exact answer using a

14 lim x 3 2 x 1 X 1

4 lim x 2 x2 5x 6 x 2

11 lim x 0 x 3 3

x 1 7 lim x 1x 1

8 lim X 2 x 3x 2 x 4

2 lim x3 2x 3x 3 x 1

1 lim 3x 2x 1 x 2

1 Find T N B K at t ln 2 if r 2t e 4pt s 2 Draw the curve C of r 2pt s 3 On C draw T N and the circle of curvature at t In 2 2p

Determine the area of the shaded region in the figure 10 O C S x 15 15 dx What is the area of the region in terms of y 0 5 OA 2y dy 1 15 dy 12v 0y 3 0 5 OC 15 2y dy 3 5 2 The area of the shaded region is 2y 15 dy x y 15 x 2y 15 D 2 B 6 x 15 dx 15 5 3 10 10 x 16 2 X S dx 6 2y y 15 dy 15 OD 2y y 15 dy 24

Perform 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 a Locate the critical points of the given function Select the correct choice below and if necessary fill in the answer box to complete your choice A The critical point s is are at x Simplify your answer Use a comma to separate answers as needed OB The function does not have a critical point b Locate the minimum value Select the correct choice below and if necessary fill in the answer box within your choice A There is a local minimum at x Simplify your answer B There is no local minimum OA 52 Locate the maximum value Select the correct choice below and if necessary fill the answer box within your choice There is a local maximum at x Simplify your answer OB There is no local maximum 5 3 2 4

Determine the area of the shaded region in the 6 10 I V x 15 dx x 15 2 dx 15 6 OA O C 10 x 15 2 dx 15 What is the area of the region in terms of y 0 OA 2yc 3 5 5 j 1 16 2y dy dy OC x 15 2y dy The area of the shaded region is Type an exact answer x y 2 15 x 2y 10 15 OB 2 x 15 d dx 10 1 6 6 D 2 2 x 15 dx x 15 15 B 2y 15 dy 3 10 OD 2y 15 dy 15 dx

Perform 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 exi

Express the area of the following shaded region in terms of a one or more integrals with respect to x and b one or more integrals with respect to y a What is the area of the region in terms of x 10 OA x 10x dx 5x 50 dx 5 0 5 10 OB 10x dx 5x 50 dx 5 0 5 10 OC f x 15x 50 dx 2 10 OD x 15x 50 dx 10x y 5x 50

Determine the area of the shaded region in the figure OA S 4x x 5 dx 5 1 0 S x 5 dx S 4x dx 5 1 What is the area of the region in terms of y 1 OA y 5 dy 1 0 S 1 dy 1 OC y 5 dy 4 0 S dy 1 The area of the shaded region is Type an exact answer y 4x y x 5 OU C OB OD B 5 1 5 x 5 4x dx x 5 dx 1 j x 6 2 oy dy y 5 dy 4x dx

The line y 5x b is tangent to the graph of y x at the point P a a Find P and determine b P b 1 100 10 Type an ordered pair Type an Integer or simplified fraction

Find the area of the shaded region shown in the graph CEL 20 y 20 sinx y 20 cos x KIN

Consider the curve y f x shown below Find f 7 and f 7 V y f x S y x 2 tangent line

Apply the three step method to compute the derivative of f x 2x f x

Compute 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

Find 1 dy if y x 12 dx dy dx Type an exact answer

Find 9 10 d dx d x 10 dx Type an exact answer

a Find the point on the curve y x where the tangent line is parallel to the line y 10 X X b On the same axes plot the curve y x the line y and the tangent line to y x that is parallel to y 10 10 3 X a The point on the curve y x where the tangent line is parallel to the line y 10 is Type an ordered pair

1 Use the power rule to find the derivative of f x 12 f x

Here is a graph of f x x and the line tangent to f x at a f a The equation of the tangent line is y 2x 1 Find a f a and the slope of the parabola f x at a f a a 1 GESED

If f x x compute f 3 and f 3 f 3

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

Find the slope of the curve y x at x 5 The slope of the curve at x 5 is f 5

1 Find the derivative of f x x8 at x 2 1 5 Simplify your answer

13 Use the power rule to find the derivative of f x x

Use the power rule to find the derivative of f x x

Find the point s on the graph to the right where the slope is equal to 6 11 The point s on the given graph at which the slope is equal to 11 is are 2 2 11 11 1 y x3 x y X Slope is 3x E 2 11 ad Uco a comma to separate answers as needed

rt the following degree measure to radian measure 33 radians Type an integer or a decimal nearest hundredth

Find the measure of each angle 5x 4 3x 6 Simplify your answers Type an integer or a fraction 5x 4 3x 6

The rotation of the smaller wheel in the figure causes the larger wheel to rotate Through how many degrees will the larger wheel rotate if the smaller one rotates through 270 3 The larger wheel rotates through approximately Do not round until the final answer Then round to the nearest integer as needed 7 2 cm