Calculus Questions

5x 1 An equation of the line tangent to the graph of y 3 1 at the point where x 1 is A 9x 7y 53 B 2x 3y 13 O C 16x 13y 44 OD 7x y 0 E x 2y 1

The family of curves represented on this graph would appear to be consistent with which of the following differential equations A d B D da Y Te tan x y d x y x x 3 E d 3x

Let f be the function defined by f x X Which of the following statements is true OA fis discontinuous at x 0 B fis an odd function O c f 0 0 OD f x 0 for x 0 E fhas a relative maximum x for for x 0 x 0

Consider the position function s t 4 9t 25t 25 Complete the following table with the appropriate average velocities and then make a conjecture about the value of the instantaneous velocity at t 3 Complete the following table Type exact answers Type integers or decimals Time Interval Average Velocity 3 4 3 3 1 3 3 01 3 3 001 3 3 0001 GOLA

The maximum value of f x e x is OA 1 OB 1 OC 2 OD In 2 OE O

For the following function make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at the indicated point f x 11 cos x at x RIN x 2 Complete the table below Round the final answer to three decimal places as needed Round all intermediate values to four decimal places as needed Slope of secant line Interval T

For the position function s t instantaneous velocity at t 0 Time Interval Average Velocity 0 1 Complete the following table Time Interval 18 t 1 0 1 complete the following table with the appropriate average velocities Then make a conjecture about the value of th 0 0 5 0 0 5 0 0 1 12 0 0 1 0 0 01 0 0 01 Average 9 16 4 Velocity Type an integer or decimal rounded to three decimal places as needed 17 8 0 0 001 0 0 001 17 9

The graph of y g x is shown below At which point is g g OA A B B O C C D D OE E LA B C D Click here for long description x

Which of the following differential equations is are not separable 1 t yt 1 dt II d y dx 3xy III dy y yx da O A I only O B II only O C III only D II and III dy dx OE I and III

13 16 Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a 13 f x 3x x 2 5 a 14 g t t 5t 2t 1 a 2

Factor completely Then solve p w 3w 6w 24w

9 Discontinuities at 0 and 3 but continuous from the right at 0 and from the left at 3

A rock is dropped off the edge of a cliff and its distance s in feet from the top of the cliff after t seconds is s t 16t Assume the distance from the top of the cliff to the water below is 256 ft Complete parts a and b below Time Interval Average Velocity its 0 of 1 a When will the rock strike the water The rock will strike the water at 4 seconds Simplify your answer b Let t a be the time when the rock strikes the water Make a table of average velocities and approximate the velocity at which the rock strikes the water a 1 a a 0 5 a a 0 1 a a 0 01 a a 0 001 a ft sec Type an integer or decimal rounded to three decimal places as needed Save

a Graph the function f x x 4x 3 b Identify the point a f a at which the function has a tangent line with zero slope c Confirm your answer to part b by making a table of slopes of secant lines to approximate the slope of the tangent line at this point

Consider the position function s t 4 9t 291 17 Complete the following table with the appropriate average velocities and then make a conjecture about the value of the instantaneous velocity t 4 Complete the following table Type exact answers Type integers or decimals Time Interval Average Velocity 4 5 4 4 1 4 4 01 4 4 001 4 4 0001 15 1 10 69 10 249 10 2049 10 2005

If the graph of y A 5 B 1 asymptote x 3 then a c C 5 D 1 E 0 ax b x c has a horizontal asymptote y 2 and a vertical

If x 2xy y 2 at the point 1 1 dy is da A 0 1 OB B 2 O C 12 2 OD nonexistent OE

The differential equation dy dx O A y Cekt OB y C x 4 O C y Ce x O D y e C E x y C af has a solution given by

Which of the following is not a differential equation A y 0 dx OB 4 y 2x dy c x x dx 4 y Y O D t 1 2 4 BOULL d y dt2 2 E t s ds

The change in the momentum of an object Ap is given by the force F acting on the object multiplied by the time interval that the force was acting Ap FAt If the force in newtons acting on a particular object is given by F t cost what s the total change in momentum of the object from time t 5 until t 7 seconds A 0 708 newton sec B 0 909 newton sec C 1 616 newtons sec D 0 402 newton sec E 1 416 newtons sec

A separable differential equation is a first order differential equation that can be algebraically manipulated to look like A Both f y dy g x dx and f x dx f y dy B f x dx f y dy O c f y dy g x dr O D f x dx f y dy O E g y dx f x dx

If y5 3x y 5x4 49 then at the point 1 2 is dy dx O A 11 23 OB 11 23 O C 1 O D OE O 10 3

A solid has as its base the circular region in the xy plane bounded by the graph of x y2 4 Find the volume of the solid if every cross section by a plane perpendicular to the x axis is a quarter circle with one of its radii in the base 32 A 32T OA 32 B 3 TT C 8T OD 3 E 138 T 5 OE

If y5 3x y 5x4 49 then dy at the point 1 2 is dx O A OB 11 23 O C 1 11 O D OE O D 100 3

limx 9x x 7x4 x 12x A 9 OB 7 O C C 72 12 OD 7 O E Does not exist

Suppose a certain country s population has constant relative birth and death rates of 97 births per thousand people per year and 47 deaths per thousand people per year respectively Assume also that approximately 30000 people emigrate leave from the country every year Which of the following equations best models the population P P t of the country where tis in years O A 50P 30000 dP dt dP B dt 05P 30000 dP O C 5P 30000 dt OD O E d 05P 30000 5P 30

The temperature over a given period is Estimate the average temperature from 0 t 8 using the left endpoints of four equal subintervals Time hours 0 1 2 3 4 5 6 7 8 Temp F 35 38 45 40 37 22 14 8 13 A 26 4 F B 26 F C 50 F OD 32 75 F

OA ye dy xdx O B ydy xe d C None of these The equation is not separable OD ydy xe dx E udy re 2 dr

The area of the region in the first quadrant between the graph of y k 4x x 0 and x axis where k 0 is OA KT B 2 k OC 2kT D k O D O E 2

dx ves the slope at any point on the graph of the f 2 8 A Write the equation of the tangent line to f x at point 2 8 B Write an expression for f x in terms of x C What is the domain of f x D What is the minimum value of f x What is the minimum E Using the axes provided sketch a slope field for the given differential equation at the nine point indicated

Which of the following graphs is consistent with the differential equation dy dz 2 O A B C D E Vi

is increasing at a rate directly proportional to 3500 C t Also C 0 1000 and C 5 2000 A Find C t as a function of t only B Calculate C 10 C Find lim C r and explain its meaning 1 w D On the axes provided draw a graph showing the number of cougars as a function of time 4000 3000 2000

X 0 9 1 0 1 1 Given h x 5 3 5 0 4 7 evaluated at x 5 O A 3 OB 3 O C 1 O D OE 3 what is a reasonable value forh x

Consider the following graph of y f x 40T 2 20 0 20 40 60 2 4 O A I only OB II only O C III only OD II and III only E I II and III 6 8 10 Which of the following could be the equation for f x I A Ba with 0 a 1 B 0 and A 0 I A Ba 2 with a 1 B 0 and A 0 III A a B with 0 a 1 B 0 and A 0

Let f be a polynomial function with degree greater than 2 If a b and f a f b 1 which of the following must be true for at least one value of x between a and b I f x 0 11 f x 0 III f x 0 OA None OB I only O C II only OD I and II only OE I II and III

The average value of the function g x 3cos on the closed interval T is OA 7 593 OB 4 347 O C 1 325 OD 18 068 OE 30 980

Which of the following is a separable first order differential equation O A y 2 In xy x y OB d x Y O c dy da xy x y dy O D sin x O E y x y 2x

If f 2 2 5 and f 2 2 5 then f 2 5 is approximately OA 2 5 OB 2 OC 2 5 OD 1 25 OE 2

Consider the position function s t 4 9t 29t 17 Complete the following table with the appropriate average velocities and then make a conjecture about the value of the instantaneous velocity at t 4 Complete the following table Type exact answers Type integers or decimals Time Interval Average Velocity 4 5 4 4 1 4 4 01 4 4 001 4 4 0001

Jo 1 4 O A In 1 4 O B jarctan 24 O C j arcsin 22 O D In 1 x4 O E arctan z

5 6 The graph of a function f is given a At what numbers a does limx a f x not exist b At what numbers a is f not continuous c At what numbers does lim f x exist but is not continuous at a n 5 e 6

A rock is dropped off the edge of a cliff and its distances in feet from the top of the cliff after t seconds is s t 16t Assume the distance from the top of the cliff to the water below is 256 ft Complete parts a and b below a When will the rock strike the water The rock will strike the water at seconds Simplify your answer

The slope of the line normal to the curve ex x 2 10 at the point 0 3 is OA O B 6 O C 6 O D 6 OE Undefined

Consider the position function s t 16t 104t representing the position of an object moving vertically along a line Sketch a graph of s with the secant line passing through 0 75 s 0 75 and 2 s 2 Determine the slope of the secant line and explain its relationship to the moving object A As 144 69 0 75 Q OB AS 144 69 0 75 2 ON The slope of the secant line is msec 60 Type an integer or a decimal What is the relationship of the slope to the moving object OA The slope is the average velocity of the object over the interval 69 144 OB The slope is the instantaneous velocity when t 2 C The slope is the maximum velocity of the object OD The slope is the average velocity of the object over the interval 0 75 2 144 69 0 75 2 G

3 Write the equation of a vertical line passing through 8 5

2 Which of the following equations is perpendicular to the graph of 4x 3y 10 A 3x 4y 10 B 4x 3y 10 C 3x 4y 12 1 D 3y 12 4

3 6 A linear function contains the points 2 5 and 15 20 Does the point 43 53 also lie on the line Explain 2

4 Line 1 passes through the points 2 5 and 6 k and is perpendicular to the line given by y 1 x 1 5 Find the value of k

The graph of f x is shown below along with the line y x C a y x y f x d Click here for long description O D d f x 0 da OE f c a OA f x is a one to one function B a f f a C f x has a domain of d x C Assuming that f x is defined by restricting f x to the domain a x b which of the following statements is false b

1 Write the equation of the line that is perpendicular to y 5x and passes through 7 2