Calculus Questions

Find the average rate of change of the following functions over the given interval a f x 3x 7 over the interval 0 3 b g x 2 over the interval x x h

2 Let f be twice differentiable on R and suppose f x 0 for all x ER a Show that f is monotonic on R i e that f is increasing on all of R or decreasing on all of R b Show that f is increasing if and only if f 1 is increasing Note To show this if and only if you must show two implications if f is increasing then f is increasing and if f 1 is increasing then f is increasing c Show that if f is increasing and concave up everywhere then f 1 is increasing and concave down everywhere

1 Assume S is a solid in three space lying entirely between the planes x a and x b each of whose x cross sections S is a circle of radius r x where r is a continuous function Fix positive scaling factors a and 6 and let T be the solid whose points are of the form ax By Bz where P x y z is a point of S In other words T is obtained by stretching S by an x scaling factor of a in the x direction and by a yz scaling factor of in the y and z directions a Using the volume via cross sections definition derive a formula for vol T in terms of vol S a and B Hint describe the x cross sections T in terms of the x cross sections of S What kind of shapes are they How do their dimensions relate to the dimensions of the cross sections of S Suggestion it may help to do the sketch in part b to help you visualize what is going on b Consider the particular example where S is the solid sphere of radius 1 centered at the origin and T is the solid obtained by stretching S as above with the x scaling factor a 4 and the yz scaling factor 2 i Provide a sketch of T that includes x y and z axes with scales Hint start with 6 easy to plot points on the sphere S and determine what happens to them when you stretch by c and d Computo

Use implicit differentiation to find the equation of the tangent line to the curve xy xy 4 at the point 2 1 The slope of this tangent line is m The equation of the tangent line is y

If 1 x 1 y 1 and y 4 find y 4 by implicit differentiation

Solve the given equation Enter your answers as a comma separated list Let k be any integer Round terms to three decimal places where appropriate If there is no solution enter NO SOLUT tan 8 9 2 cos 0 1 0 8

Use the graph shown to find the following a The domain and range of the function b The intercepts if any c Horizontal asymptotes if any d Vertical asymptotes if any e Oblique asymptotes if any A The function has two horizontal asymptotes The top asymptote is Type equations GLICE C Find the nozomial asymptotes I were are any Select the confect choice below and if necessary we answer boxes to complete your choice A The function has one vertical asymptote Type an equation and the bottom asymptote is Q B The function has one horizontal asymptote y 3 Type an equation C The function has no horizontal asymptotes d Find the vertical asymptotes if there are any Select the correct choice below and if necessary fill in the answer box es to complete your choice OB The function has two vertical asymptotes The leftmost asymptote is and the rightmost asymptote is Type equations OC The function has no vertical asymptotes

olve the given equation Enter your answers as a comma separated list Let k be any integer Round terms to three decimal places where appropriate If there is no solution enter NO SOLUTIO 3 sin 0 7 sin 8 2 0 2 80 k x N

Solve the given equation Enter your answers as a comma separated list Let k be any integer Round terms to three decimal places where appropriate If there is no solution enter NO SOLUTI tan 0 9 2 cos 8 1 0 e

Find the vertical horizontal and oblique asymptotes if any of the following rational function 5x 17x 12 2x 7x 4 Q x Find the vertical asymptotes Select the correct choice below and if necessary fill in the answer box es to complete your choice 1 The function has one vertical asymptote x 2 Type an equation Use integers or fractions for any numbers in the equation OB The function has two vertical asymptotes The leftmost asymptote is and the rightmost asymptote is Type equations Use integers or fractions for any numbers in the equations OC The function has no vertical asymptote Find the horizontal asymptotes Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The function has one horizontal asymptote Type an equation Use integers or fractions for any numbers in the equation OB The function has two horizontal asymptotes The leftmost asymptote is and the rightmost asymptote is Type equations Use integers or fractions for any numbers in the equations OC The function has no horizontal asymptote

Find u v U V u i 4j v 6i j

Solve the given equation Enter your answers as a comma separated list Let k be any integer Round terms to two decimal places where appropria cos 8 0 24 0 rad List six specific solutions 8 rad

1 Points DETAILS Solve the given equation Enter your answers as a comma separated list Let k be any integer Round terms to three decimal places where appropriate If there is no solution enter NO SOLU sin 0 2 cos 8 1 0 0

1 For the function F x 5 asymptotes 1970 a graph the rational function using transformations b use the final graph to find the domain and range and c use the final graph to list any vertical horizontal or oblique X B There is one vertical asymptote It is x 0 Type an equation Use integers or fractions for any numbers in the equation C There is no vertical asymptote What is are the horizontal asymptote s Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The top horizontal asymptote is The bottom horizontal asymptote is Type equations Use integers or fractions for any numbers in the equations B There is one horizontal asymptote It is y 5 Type an equation Use integers or fractions for any numbers in the equation OC There is no horizontal asymptote What is are the oblique asymptote s Select the correct choice below and if necessary fill in the answer box es to complete your choice 0 67 of 1 O A There is one oblique asymptote It is Type an equation Use integers or fractions for any numbers in the equation OB The oblique asymptote with the positive slope is and the oblique asymptote with the negative slope is Type equations Use integers or fractions for any numbers in the equations C There is no oblique asymptote Sa

Solve the given equation Enter your answers as a comma separated list Let k be any integer Round terms to three decimal places where appropriate If there is no solution enter NO SOLUTIC 3 sin 8 7 sin 8 2 0 A

Find the vertical horizontal and oblique asymptotes if any for the following rational function R x X 2 Find the vertical asymptotes Select the correct choice below and if necessary fill in the answer box es to complete your choice A The function has one vertical asymptote x 2 Type an equation Use integers or fractions for any numbers in the equation OB The function has two vertical asymptotes The leftmost asymptote is and the rightmost asymptote is Type equations Use integers or fractions for any numbers in the equations OC The function has no vertical asymptote Find the horizontal asymptotes Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The function has one horizontal asymptote Type an equation Use integers or fractions for any numbers in the equation and the bottom asymptote is OB The function has two horizontal asymptotes The top asymptote is Type equations Use integers or fractions for any numbers in the equations OC The function has no horizontal asymptote

Solve the given equation Enter your answers as a comma separated list Let k be any integer Round terms to three decimal places where appropriate If there is no solution enter NO SOLUTION cos 8 cos 8 6 0 8

Solve the given equation Enter your answers as a comma separated list Let k be any integer Round terms to three decimal places where appropriate If there is no solution enter NO SOLUTION sin 8 2 cos 8 1 0 0

Find the vertical horizontal and oblique asymptotes if any of the given rational function x 8 X 7x 10 H X 2 Find the vertical asymptotes Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The function has one vertical asymptote Type an equation Use integers or fractions for any numbers in the equation OB The function has two vertical asymptotes The leftmost asymptote is and the rightmost asymptote is Type equations Use integers or fractions for any numbers in the equations OC The function has no vertical asymptote

K 1 For the function F x 5 a graph the rational function using transformations b use the final graph to find the domain and range and c use the final graph to list any vertical horizontal or oblie X asymptotes ya moyvi vincunucuon VJU LU supru un www OC The range of the given function is yly is a real number y Type an integer or a simplified fraction OD The range of the given function is the set of all real numbers c What is are the vertical asymptote s Select the correct choice below and if necessary fill in the answer box es to complete your choice A The left vertical asymptote is The right vertical asymptote is Type equations Use integers or fractions for any numbers in the equations B There is one vertical asymptote It is x 0 Type an equation Use integers or fractions for any numbers in the equation Points 0 5 of 1 OC There is no vertical asymptote What is are the horizontal asymptote s Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The top horizontal asymptote is The bottom horizontal asymptote is Type equations Use integers or fractions for any numbers in the equations OB There is one horizontal asymptote It is Type an equation Use integers or fractions for any numbers in the equation OC There is no horizontal asymptote

A random variable X is uniformly distributed between a 3 and b 13 as shown below 2 a 3 4 5 6 7 8 9 10 11 12 13 14 Find the height h of the rectangle that corresponds to the sample space 1 Recall that if X is uniformly distributed from a to b then h b a

1 For the function F x 5 a graph the rational function using transformations b use the final graph to find the domain and range and c use the final graph to list any vertical horizontal or oblique X asymptotes 1ype an integer or a simpiimea traction OD The domain of the given function is the set of all real numbers What is the range of the given function Select the correct choice below and if necessary fill in the answer box to complete your choice OA The range of the given function is yly is a real number y Type an integer or a simplified fraction B The range of the given function is yly is a real number y Type an integer or a simplified fraction Use a comma to separate answers as needed OC The range of the given function is yly is a real number y Type an integer or a simplified fraction D The range of the given function is the set of all real numbers c What is are the vertical asymptote s Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The left vertical asymptote is The right vertical asymptote is Type equations Use integers or fractions for any numbers in the equations OB There is one vertical asymptote It is Type an equation Use integers or fractions for any numbers in the equation There is no vertical asymptote

For the continuous distribution below what does the shaded area represent 74 OP x 7 OP x 0 OP 6 x 7 0 1 2 3 4 5 6 7 8 9 9 10 X

Find all solutions of the given equation Enter your answers as a comma separated list Let k be any integer Round terms to two decimal places where appropria cos 0 1 0 0 rad

Solve the given equation Enter your answers as a comma separated list Let k be any integer Round terms to two decimal places where appropriate cos 8 1 4 0 1 32 rad

1 For the function F x 5 a graph the rational function using transformations b use the final graph to find the domain and range and c use the final graph to list any vertical horizontal or oblique X asymptotes A The domain of the given function is xlx is a real number x Type an integer or a simplified fraction B The domain of the given function is xlx is a real number x 0 Type an integer or a simplified fraction Use a comma to separate answers as needed C The domain of the given function is xlx is a real number x Type an integer or a simplified fraction D The domain of the given function is the set of all real numbers What is the range of the given function Select the correct choice below and if necessary fill in the answer box to complete your choice OA The range of the given function is yly is a real number y Type an integer or a simplified fraction OB The range of the given function is yly is a real number y Type an integer or a simplified fraction Use a comma to separate answers as needed OC The range of the given function is yly is a real number y Type an integer or a simplified fraction D The range of the given function is the set of all real numbers

Solve the given equation Enter your answers as a comma separated list Let k be any integer Round terms to two decimal places where appropriate cos 8 rad

Compute the directional derivative of the function g x y sin x x y at the point P 3 2 in the direct 8 15 17 197 Be sure to use a unit vector for the direction vector The directional derivative is IMOK

Consider the initial value problem x 1 y y y ex y 1 1 y 1 1 Does the Existence and Uniqueness Theorem guarantee the existence of a unique solution to this initial v problem Justify your answer DO NOT SOLVE THE DIFFERENTIAL EQUATION

The function t f s represents the number of school busses t needed to transport s students to school Interpret the meaning of the statement f 33 650

6 Consider the function c f d d 5 a What is the independent variable of f b What is the dependent variable of f c Find f

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

Let m f t represent the total money earned while working at a company for t years Explain what information is given by the function f Be sure to include both the input and the output in your explanation

Suppose that Find an equation for the tangent line to the graph of fat x 1 Tangent line y f x 3x 32x

8 Let g x 2x 6 a Find g b Use compositions to verify that g and g are in fact inve

2 Let f x 1 x 11 Write an 3 equation for g x the inverse of f

1 For the function F x 5 a graph the rational function using transformations b use the final graph to find the domain and range and c use the final graph to list any vertical horizontal or oblique asymptotes X OA The domain of the given function is xlx is a real number x Type an integer or a simplified fraction B The domain of the given function is xlx is a real number x 0 Type an integer or a simplified fraction Use a comma to separate answers as needed C The domain of the given function is x x is a real number x Type an integer or a simplified fraction OD The domain of the given function is the set of all real numbers What is the range of the given function Select the correct choice below and if necessary fill in the answer box to complete your choice OA The range of the given function is yly is a real number y Type an integer or a simplified fraction B The range of the given function is yly is a real number y Type an integer or a simplified fraction Use a comma to separate answers as needed OC The range of the given function is yly is a real number y Type an integer or a simplified fraction OD The range of the given function is the set of all real numbers

5 If 9 4 is on the graph of f which of the following ordered pairs is on the graph of f A 9 4 B 9 4 C 4 9 D 4 9

1 For the function F x 5 asymptotes a Choose the correct graph below OA X a graph the rational function using transformations b use the final graph to find the domain and range and c use the final graph to list any vertical horizontal or oblique X 20 B 201 OB The domain of the given function is xlx is a real number x Type an integer or a simplified fraction Use a comma to separate answers as needed b What is the domain of the given function Select the correct choice below and if necessary fill in the answer box to complete your choice OA The domain of the given function is xlx is a real number x Type an integer or a simplified fraction OC The domain of the given function is xlx is a real number x Type an integer or a simplified fraction domain of the given function is the set of all real numbers O C D Ay Q Check

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

Michelle got a new job through the McCartney Employment Agency The job pays 49 400 per year and the agency fee is equal to 35 of one month s pay How much must Michelle pay the agency Round to the nearest cent

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

f x x 10x 9 9 following polynomial function Write f in factored form The complex zeros of fare i i 3i 3i Simplify your answer Type an exact answer using radicals and i as needed Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed Use the complex zeros to factor f f x Factor completely Type an exact answer using radicals and i as needed Use integers or fractions for any numbers in the expression 4

Solve the equation Check for possible extraneous solutions log x 6 loge 3 2 x 6 X 5 no solution x 2

In Exercises 71 82 find the domain of the function 71 f x 5x 2x 1 72 g x 1 2x 4 73 h t 75 77 g x g y y 1 79 f s 81 f x X 10 3 x 2 x 4 x s 1 S 4 3y y 5 76 f t 3 t 4 10 74 s y 78 h x 80 f x 82 f x x 2x x 6 6 x x 2 x 10

a Applying the two step Adam Bashforth method see Lec 8 to approximate y 2 5 where y is the solution to y 1 y 1 t 1 5 y 1 1 with h 0 5 You may use Euler s method to get the starting value at t 1 5 exact solution is y t 2t 1 b Compare your approximation with the exact solution write down your observation and try to explain why very briefly optional

4 Use the graph given for each problem to determine the values If the value is between two integers approximate to one decimal place a f 0 b f 1 c If f x 2 then x d If f x 0 then the possible values of x are x x or x or x or x 1 2 1 0

Find the following limits lim 00 jum b lim 6x5 1 7x4 4x 1 6x5 1 7x4 4x 1 II 8 8

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

Given the function f x x e7 Use the derivatives to determine the open interval s where the function is concave up algebraically Leave answers in exact form Determine the open interval s where the function is concave down Determine any points of inflection