Calculus Questions

Thinking 7 Differentiate the following without simplifying f x 4 cos x sin cosx

3 Differentiate each of the following Do not simplify a f x x cos 3x 9x b f x sinx tan x

6 Suppose the monthly revenue in thousands of dollars for the sale of xhundred units of an electronic item is given by the function R x 40x e 0 4x 30 Determine the number of units to produce in order to maximize revenue

1 Determine the derivative with respect to x Simplify fully a f x 12 4x c f x e6xsin 5x b y 1 5e x 2 d f x sin esinx

5 Find the slope and equation of the tangent line to the curve f x sin x cos x at x

4 Determine the equation of the line that is tangent to the function f x 2sinx sin x at the point x 7

2 Determine the derivative of y 2x 3 3 8x Do not simplify

Follow the steps for graphing a rational function to graph the function R x ww 3x 3 5x 25 Simplify your answer Type an integer or a fraction OC The graph has x intercept s and no y intercept Simplify your answer Type an integer or a fraction Use a comma to separate answers as needed Type each answer only once OD The graph has neither x intercepts nor y intercepts Determine the behavior of the graph of R at any x intercepts Select the correct choice below and if necessary fill in answer box es to complete your choice O A The graph will cross the x axis at x and touch but not cross the x axis at x Type integers or simplified fractions Use a comma to separate answers as needed Type each answer only once OB The graph will cross the x axis at x Type an integer or a simplified fraction Use a comma to separate answers as needed Type each answer once x intercent OC The graph will touch but not cross the x axis at x Type an integer or simplified fraction Use a comma to separate answers as needed Type each answer of once

2 Jypu un muyor ora B xx Type an integer or a simplified fraction C x x 5 Type an integer or a simplified fraction Use a comma to separate answers as needed D The domain is the set of all real numbers Write R x in lowest terms Select the correct choice below and if necessary fill in the answer box to complete your choice A R x Simplify your answer B R x is already in lowest terms Locate the intercept s of the graph Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The graph has x intercept s and y intercept Simplify your answers Type integers or fractions Use a comma to separate answers as needed Type each answer only once OB The graph has y intercept and no x intercept Simplify your answer Type an integer or a fraction OC The graph has x intercept s and no y intercept Simplify your answer Type an integer or a fraction Use a comma to separate answers as needed Type each answer only once OD The graph has neither x intercepts nor y intercepts

Problemy 1 Use left or right Riemann sums with 4 rectangles to estimate the following areas 1 f x x from 0 2

1 4 Verify by differentiation that the formula is correct 1 fln x dx x lnix x C 2 f tan x dx tan x x C 3 S 1 x 1 x dx 4 x a bx dx S 1 x 2 1562 3bx C 2a a bx 2 C

1 4 Verify by differentiation that the formula is correct 1 f In x dx x lnix x C 2 tan x dx tan x dx tan x x C

25 54 Evaluate the integral 3 25 x 2x 4 dx 27 1 3t dt 29 x dx 31 33 0 1 1 dt cos e de 2 26 28 30 32 S 1 0 100 X dx 1 8v 16 fx 2 3 dx 3 1 L dz 2 3 5 34 Se dx

4 Imagine you are driving a car on a highway and your distances in meters from a certain point is given by the function s t 30t 2t where t is the time in seconds since you passed that point a Determine the time at which you pass the point for the second time b Calculate your average speed over the first 10 seconds after you pass the point for the first time c Find the limit of your average speed as t approaches 0 Interpret the meaning of this limit in the context of the problem Hint Average speed over an interval a b is given by s b s a b a

25 54 Evaluate the integral 25 x 2x 4 dx 27 31 1 31 dt 29 S x dx 26 fx100 dx 28 1 8v 16v du f 10 30 S 2 3 dx

f x 1 sin if x 0 if x 0 0 a Prove that 1 f x 1 for all x b Determine the value of lim o f x if it exists If it does not e explain why

j limx 2 2 x2 421 8 m lim 0 cos r 1 x Hin k lim 4 4 n lim 2 32 10 x 4x 4 www 3r 2r 5 1 limr 4 6 r 3x 2 o lim 1 2 3 x

Solve the rational inequality graphically using a graphing calculator Give answers in interval notation If necessary round your answer to the nearest tenth 0 x 1 x x 5

3 Consider the function k x r sin 1 Show that 0 k r x for a x 0 Find lim o k r using the Squeeze Theorem

Follow the steps for graphing a rational function to graph the function P x x x 6 36 Determine the vertical asymptote s if one exists 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 Type an equation Use integers or fractions for any numbers in the equation B The function has two vertical asymptotes The leftmost asymptote is x 6 and the rightmost asymptote is x 6 Type equations Use integers or fractions for any numbers in the equations the middle asymptote is and the rightmost asymptote is OC The function has three vertical asymptotes The leftmost asymptote is Type equations Use integers or fractions for any numbers in the equations D The function has no vertical asymptote Determine the behavior of the graph on either side of any vertical asymptotes if one exists Select the correct choice below and if necessary fill in the answer box es to complete your E and on the other It approaches either co or co on both sides of the asymptote s at x OA It approaches o on one side of the asymptote s at x Type integers or simplified fractions Use a comma to separate answers as needed Type each answer only once OB It approaches either or on both sides of the asymptote s at x Type an integer or a simplified fraction Use a comma to separate answers as needed Type each answer only once and o on the other OC It approaches oo on one side of the asymptote s at x Type an integer or a simplified fraction Use a comma to separate answers as needed Type each answer only once D The function has no vertical asymptote

25 54 Evaluate the integral S x 2x 4 dx 25 26 S x 100 dx

e lim o tan x I 2x 3x 1 f lim 1x 37 2r 5

A company determined that the marginal cost C x of producing the xth unit of a product is given by C x x 4 Find the total cost function C assuming that C x is in dollars and that fixed costs are 6000 C x

Differentiate y xe 7x 8e x x8 Choose the correct answer below O A dy 7xe 7x e 7x 8e x 8x dx OB dy dx OC dy dx OD dy 7xe 7x e 7x 8e x 8x 7xe 7x e 7x 8e X 8x 7xe 7x e 7x 8e 8x dx

Differentiate y 12 log 5x d 12 log 5x dx

Find the integral 9 X 9 10 10 6 X dx dx

A company s total cost in millions of dollars is given by C t 130 60e where t is the time in years since the start up date The graph of C t is shown to the right Find each of the following a The marginal cost C t b C 0 c C 3 d Find lim C t and lim C t 10 t Do not include the symbol in your answer a C t b C 0 million per year Simplify your answer Do not include the symbol in your answer million d lim C t 1 Simplify your answer Do not include the symbol in your answer lim C t 118 per year wor Do not include the symbol in your answer Ay 150 75 0 c C 3 per year Simplify your answer Round to the nearest thousand as needed Do not include the symbol in your answer 0 15 30 Q Q

erentiate y 340 48 3 8 340

Solve the following integral The answer can be determined by rules of antidifferentiation but some algebra may required beforehand 7 4u5 8 u 4u5 8 du du

Find the integral 10 S 77 Ox dx

Find s t where s t represents the position function v t represents the velocity function and a t represents the acceleration function a t 6t 8 with v 0 5 and s 0 8 s t

10p b lim x 0 sin 3x cot 5x x cot 4x Do not use L Hopital s Rule

Find the equation of the line segment between Po 3 1 4 and P 0 5 2

5 Find the area of the a u 2ij b u 8i 2j 3k v parallelogram that has two adjacent sides u and v 2k v 3i 2j k 2i 4j 4k

Suppose that a particle following the path c t t t 5t 0 flies off on a tangent at to 12 10 Compute the position of the particle at the time t Enter your answer in the vector form Use symbolic notation and fractions where needed

A pendulum of length 10 cm satisfies the equation 0 0 0 The bob is released from a starting angle 80 8 Find an equation for the position at any time and find the per of the motion g 9 8m s o k0 0 1 8 T T RT B

Section 4 2 Prove that if f is increasing then f a to illustrate this property f b f a b a for all ab Draw a picture

Find the inflection point for the function shown below If there are more than one be sure to spearate them by using a comma If there is not an inflection point record DNE in the answer box If necessary round all numbers to two decimal places f x 2x 18x 96x 3

Solve the inequality Enter your answer using interval notation x 2 x 3 X 1 X 2

Solve the inequality Enter your answer using interval notation 1 7 1 X X 6 VI

Find the domain of the given function Enter your answer using interval notation 1 f x x482x2 81

Solve the inequality Enter your answer using interval notation 9x 49 x 25 0

Solve the inequality Enter your answer using interval notation x 4x 5 5x222x 15 0

Solve the inequality Enter your answer using interval notation x 1 x2 3 3 1 x 1

Find the global maximum and global minimum values of the function g 0 60 7 sin 0 on the interval 0 Use the derivatives to algebraically answer the question and leave answer in 4 decimal places The global minimum value The global maximum value H 0 3591 6 OF

Suppose 1100 square centimeters of material is available to make a box with a square base and an open top Find the largest possible volume of the box Give your answer rounded to the nearest whole cubic centimeter Volume 3510 57 cubic centimeters

1 point The three series An Bn and Cn have terms 1 1 n10 n5 0 1 2 3 n n 1 An Use the Limit Comparison Test to compare the following series to any of the above series For each of the series below you must enter two letters The first is the letter A B or C of the series above that it can be legally compared to with the Limit Comparison Test The second is C if the given series converges or D if it diverges So for instance if you believe the series converges and can be compared with series C above you would enter CC or if you believe it diverges and can be compared with series A you would enter AD n 1 5n n 1309n15 Bn 10 9n5 5 5n 8n 5n10 9n35 8n n 8n 9n13 5n 2 7 2 Cn 1 n Note To get full credit all answers must be correct Having all

In each part find a formula for the general term of the sequence starting with n 1 Enter the following information for an a an b an c an d 111 n 1 1 1 1 3 n 1 1 1 3 0 1 1 1 3 9 27 2 1 1 9 27 2 8 26 80 2 3 9 27 81 1 4 9 2 2 V v v 9

The graph of f x 3 is reflected across the x axis shifted downward 8 units and then shifted left 3 units Let g x represent the new function What is an equation for this function g x What is the y intercept of g x Enter the answer as an ordered pair What is the domain of g x Enter the answer as an interval What is the range of g x Enter the answer as an interval help points help intervals help intervals help formulas

For the following exercises start with the graph of f x 2 Then write a function g x that results from the given transformation Shift f x 1 units upward g x help formulas Shift f x 8 units downward g x Shift f x 5 units left g x Shift f x 4 units right g x Reflect f x about the x axis g x Reflect f x about the y axis g x B help formulas help formulas help formulas help formulas help formulas