Calculus Questions

Stan lim 7 sin x x 0 em Choose the correct answer below and if necessary fill in any answer boxes to complete your choice OA L lim 6 tan x In 7 sin x is finite and therefore lim 7 sin x Stan x 0 X 0 Type exact answers in simplified form OB can be written as the expression The limit does not exist because it has the indeterminate form 0 which cannot be written in the form 010 or O D The limit does not exist because it has the indeterminate form 1 which cannot be written in the form 10 The evaluated limit is Colo 88 or O C 0 00 The limit does not exist because it has the indeterminate form oo which cannot be written in the form so that H pital s rule can be applied or so that H pital s rule can be applied 80 88 so that Hopital s rule can be applied

limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates 0 05x xe Use the given functions to create an expression whose limit can be used to determine which function grows faster lim 818

O Points 0 of 1 se limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates 27 In x Inx Set up the functions as an expression of a limit to determine which grows faster lim X10 W

OA lim x b OC lim OC x X 00 b lim b Apply L H pital s Rule Choose the correct answer below lim lim Evaluate x 00 b lim x b X x p X x p lim x xb P X x p x p P 1 x Inb p b lim 1 lim x 00 Inb p b X lim x 00 Inb p b x p x p x p P J Simplify your answer TIM B lim lim x 00 b D lim OB OD x x 00 b lim x b xxb x p lim x wb x p lim x p x b x p lim 300 Inb p b a lim a p Clear all

Use the graph of y f x assuming f x 0 if x b or f to answer parts A through F a b C d www Ay O A a c c d e g OC b c c d e g OE d e g h OG b c c e g h 9 h B Identity intervals on which the graph of t is concave downward Choose the correct answer below O B e g OD a b e g OF a c c e g h OH The graph is never concave downward

Points 0 of 1 limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates Set up the functions as an expression of a limit to determine which grows faster lim X 00

Prove that lim x 00 OE Write the limit in the form L H pital s Rule can be used to evaluate the limit but it must be put in the form A L lim xin 818 OB L lim x In x 00 e for a 0 Oc L lim xin X 00 OD L lim x In x x lim X40 lim In 1 a x 1 x X 8 In 1 lim In 1 a x 840 lim xx X In 1 a x 1 In 1 ax olo or 88 How should the limit be rewritten Apply l H pital s Rule by finding the derivative of the numerator and the denominator separately In 1 a x lim

Let R be the region bounded by the following curves Let S be the solid generated when R is revolved about the given axis If possible find the volume of S by both the disk washer and shell methods Check that your results agree y x 2 1 x 0 y 26 revolved about the y axis A 26 S H G 1 2 2 dy Type exact answers OB S dy S dy Type exact answers Set up the integral s that gives the volume of the solid as a single integral if possible using the shell method Use increasing limits of integration Select the correct choice below and fill in any answer boxes within your choice OA SO x dx s dx Type exact answers dx Type exact answers ok answer

Evaluate the following limit or explain why it does not exist lim 1 2x X 0 Select the correct choice below and if necessary fill in the answer box to complete your choice lim 1 2x X 0 Type an exact answer OB The limit does not exist because it is not defined as x 0 OC The limit does not exist because l H pital s Rule cannot be applied OD The limit does not exist because the limit approaches co as x 0 Occas OA

Use analytical methods to evaluate the following limit sin rx lim x 4 x 4 sin xx lim x 4 X 4 Simplify your answer

Use analytical methods to evaluate the following limit lim x 00 lim x48 6 x 4 4x 4x Simplify your answer

Evaluate the following limit lim 1 ax for positive constants a and b x 0 b lim 1 ax x 0 Type an exact answer

Use the graph of y f x assuming f x 0 if x b or f to answer parts A through F www 9 h Q Q A Identify intervals on which the graph of f is concave upward Choose the correct answer below OA The graph is never concave upward OC a c c d e g OE e g OG a c c e a h OB d e g h OD a b e g OF a b d e g h OH b c c e a h

A boat is towed toward a dock by a cable attached to a winch that stands 10 ft above the water level see figure Let 8 be the angle of elevation of the winch and let L be the length of the cable as the boat is towed toward the dock Complete parts a to c below b Compute de dL When L 50 ft When L 20 ft when L 50 20 and 11 ft de dL de dL 10 ft 0 0041 Round to four decimal places as needed Round to four decimal places as needed

Find the intervals on which the graph of f is concave upward the intervals on which the graph of f is concave downward and the inflection points f x In x 4x 20 A w 4 10 07 Type your answer in interval notation Type an exact answer Use a comma to separate answers as neede OB The graph is never concave downward Determine the x coordinates of any inflection points of the graph of f x Select the correct choice below and if necessary fill in the answer box to complete your choice Type an exact answer Use a comma to separate answers as needed OA X OR There are no inflection points

K Evaluate the following integral Sx 6 x 6x x

How can it be shown that f x and g x have comparable growth rates f x g x Show that lim 848 OB Show that lim 818 O C O D f x g x Show that lim Show that lim f x lim g x 848 x x X 00 M where 0 M o M is nonzero and finite f x 9 x 00 grow at a comparable rate as x 00 0 Substitute the given functions f x and g x into the limit function lim 0

Consider f x 2 x 1 sin a Graph f with a graphing utility b Compute and graph f c Verify that the zeros of f correspond to points at which f has a horizontal tangent line a Each of the graphs below is shown in a 1 1 0 2 by 2 8 1 viewing window Which is the graph of f OA OB OC oo x on the interval 1 1 S OD A

Identify the relationship between Za and Zb Show your work here a O Corresponding angles O Alternate exterior angles Same side exterior angles b O Alternate interior angles O Same side interior angles

R be the region bounded by the following curves Use the method of your choice to find the volume of the solid erated when R is revolved about the y axis x y 2x 2 x 3 and x 6 volume of the solid is 108 e an exact answer Save

The function V t 3500t 18000 where Vis value and t is time in years can be used to find the value of a large copy machine during the first 5 years of use a What is the value of the copier after 2 years and 3 months After 2 years and 3 months the copier is worth 10 125 OF b What is the salvage value of the copier if it is replaced after 5 years After 5 years the salvage value of the copier is 500 c State the domain of this function o O OF d Sketch the graph of this function 1 X 18000 16000 14000 12000 10000 8000 6000 4000 2000 Clear All Draw 2 3 t 5 e What is the range of the function 500 V t 1800

K ctly toward a 150 m skyscraper that stands on the edge of a harbor The angular size of the building is the angle formed by lines from the top and bottom of the building to the observer see figure Complete parts a and b below a Use a trigonometric equation to find an expression for the angle 0 in terms of x COS X 150 Type an equation 150 m

Evaluate the following integral 2 81z 25 dz 2 1 dz dz 81z 25

An 8 ft tall fence runs parallel to the wall of a house at a distance of 6 ft Find the length of the shortest ladder that extends from the ground to the house without touching the fence Assume the vertical wall of the house is 20 ft high and the horizontal ground extends 25 ft from the fence The length of the shortest ladder is 36 89 ft Round the final answer to the nearest tenth as needed Round all intermediate values to the nearest thousandth as needed Correct 4

Let R be the region bounded by y x x 1 and y 0 Use the shell method to find the volume of the solid generated when R is revolved about the line y 3 Set up the integral that gives the volume of the solid using the shell method Use increasing limits of integration Select the correct choice below and fill in the answer boxes to complete your choice Type exact answers OA 2n y 3 1 y dy B 0 JO The volume is dx GREED Type an exact answer

Given the following information evaluate the other five functions 5 sec 8 and 0 8 3 cos 8

OA K Use a right triangle to write the following expression as an algebraic expression Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x Hint Use sin 20 2 sin 0 cos 8 sin 2 cos1 17x Identify the right triangle that can be used to simplify the given expression Choose the correct answer below 17x a Part 2 of 2 1 17x a O C HW Score 45 65 10 5 of 23 points O Points 0 of 1 sin 2 cos 17x 3ux 1 289x Simplify your answer including any radicals Use integers or fractions for any numbers in the expression O D 4 4 17x 17x 1 a Save

7 A trough has a trapezoidal cross section with a height of 7 m and horizontal sides of width 2 m and 7 m Assume the length of the trough is 14 m See the figure to the right Complete parts a and b below OA Sdx B 7 S 68600 7 y dy 0 7m Simplify your answer Use integers or fractions for any numbers in the expression Set up the integral that gives the work required to pump the water out of the tank Use increasing limits of integration Select the correct choice below and fill in the answer boxes to complete your choice Type exact answers of work is required to pump the water out of the trough 7m 3 5 m 14 m Litho math palette as needed Round to six decimal places as needed

ne Z 43 W

Find the value of a Show your work here a 97 4a 7

Angles Q and P are a linear pair Angle Q is 49 What is the value of angle P Show your work here

If Zx and 4k are supplementary and 4k 142 solve for x Write your answer in degrees k

Which pair of angles is a linear pair 8 Show your work here B and and G 6 Fland C 4 2 8 6 4 B 2G 4 6 8 F 4 A OB and F 6 None of the above 8

Solve for the unknown angle in degrees 8421 and 8431 84318618 find are complementary angles If 8431

Given the following intersecting lines how can angles 4 and 6 be classified Show your work here Vertical Angles Adjacent Angles None of the above 1 6 5 2 3 Supplementary Angles Complementary Angles

K The figure to the right shows the shape and dimension of a small dam Assuming the water level is at the top of the dam and that the shape of the dam is an isosceles triangle find the total force on the face of the dam OA s B 2 27 Set up the integral that should be used to find the total force on the face of the dam Use increasing limits of integration Select the correct choice below and fill in the answer boxes to complete your choice Type exact answers 0 dx 2 1000 9 8 27 y Y dy The total force is C 18 m numbol in the math palette as needed Round to the nearest hundredth as needed 27 m

Solve for y Y 52

The point H is the midpoint of GI Find the length of GI G 2x 7 H 3x 12 I

Find the area of the figure The figure is not drawn to scale 14 in 6 in 16 in 7 in

Find the length of AC A 9 B

Multiply the two polynomials x 2x 3 x 2x x 4 What is the resulting expression work here

5 12 pts How many years will it take for an initial investment of 20 000 to grow to 50 000 a Assume an annual interest rate of 4 3 compounded annually b Assume an annual interest rate of 4 3 compounded continuously Give exact and approximate solutions

3 6 pts Solve the equation Give exact and approximate solutions In x 4 In 2x 5

847m8110m123m 81 6m 18115m781818 16m6 m5 7m Show your work here Hint To add an exponent r type exponent or press A

Find the product and simplify 8 2a 184 2a814 1 Show your work here Hint To add an exponent a type exponent or press A

Simplify the following expression 8 9a1810 81 819a1483 31 Show your work here Hint To add an exponent a type exponent or press A

4 6 pts The population of a city is described by the function P t 40 600 1 021 where t is the number of years after 2010 In what year will the population reach 52 500 Give exact and approximate answers

2 6 pts Solve the equation log 5 x log 2 x 2

A fence is to be built to enclose a rectangular area of 450 square feet The fence along three sides is to be made of material that costs 3 per foot The material for the fourth side costs 9 per foot Find the dimensions of the rectangl that will allow for the most economical fence to be built The short side is ft and the long side is ft

A car rental agency rents 200 cars per day at a rate of 30 per day For each 1 increase in rate 5 fewer cars are rented At what rate should the cars be rented to produce the maximum income What is the maximum income The rental agency will earn a maximum income of when it charges per day