Exam 2

Question 1

For what values of a does limr * (x) =r(a) if r is a rational function? x->a

Answer 1

Those values of a for which the denominator of the function r is not zero.

Question 2

If f(x) is a polynomial, then f(x) is continuous. You know the above statement is true. Which of the following is also true?

Answer 2

If f(x) is not continuous, then it is not a polynomial

Question 3

Is the lim = infinity and lim = -infinity.
x-> 10^- x->10^+

The limit of 10 dosent exist but what exists at 10?

Answer 3

Vertical asymptote

Question 4

Find all vertical asymptotes of the given function.

g(x) = (2x)/(x - 5)

Answer 4

x=5

Question 5

Given a function f, what does f’ represent?

Answer 5

The instantaneous rate of change at any point in the domain

Question 6

The average rate of change over the interval [a, x] is _______ the limit _________ is the slope of the _________ line; it is also the limit of average rates of change , which is the instantaneous rate of change at x = __

Answer 6

(f(x)-f(a))/(x-a)

lim ((f(x)-f(a))/(x-a)
x->a

tangent

a

Question 7

What is true about the graph of f(x) = |x| at the point (0, 0) ?

Answer 7

The graph has no tangent line.

Question 8

The function f(x) = |x| has a derivative at x = 0 .

Answer 8

False, since the limit from the left and the limit from the right are not equal, the derivative does not exist

Question 9

Water is being poured into a cylindrical vase . The height of the water changes as more water is poured in. What can be said about the instantaneous change in the height with respect to the volurne of water in the vase?

Answer 9

The instantaneous change in the height with respect to the volume of water in the vase is constant

Question 10

A slow freight train chugs along a straight track. The distance it has traveled after x hours is given by a function f(x) An engineer is walking along the top of the box cars at the rate 3.2 mi/hr in the same direction as the train is moving. What is the speed of the man relative to the ground?

Answer 10

f’(x)+3.2

Question 11

is differentiable at a, must be continuous at a?

Answer 11

Yes, if is differentiable at a, then fis continuous at a

Question 12

If f is continuous at a, must be differentiable at a?

Answer 12

No continuous at a It not necessarily true that the limit that defines f’ at a

Question 13

The derivative of e to the power of any number will always be?

Answer 13

0

Question 14

What is the maximum number of horizontal asymptotes that a function can have?

Answer 14

Two

Question 15

True or False. A function can cross its horizontal asymptote

Answer 15

True

Question 16

lim e^x

x->infinity

Answer 16

Infinity

Question 17

lim e^x

x->-infinity

Answer 17

0

Question 18

lim e^-x

x->infinity

Answer 18

0

Question 19

lim (sin(7x))/19x

x->infinity

Answer 19

0

Question 20

lim 3x^25

x->-infinity

Answer 20

Look to see if powers is even or odd

Answer= -infinity

Question 21

limx^-7

x->infinity

Answer 21

0

Question 22

Determine the end behavior

f(x) = sin(5x)

For positive and negative infinity

Answer 22

Both do not exist

Question 23

Give the three conditions that must be satisfied by a function to be continuous at a point

Answer 23

lim f(x) = f(a) 
x->a

f(a) is defined

lim x a f(x) exists
x->a

Question 24

Determine the interval(s) on which the function f(x) = e ^ sqrt(x)

Is continuous then evaluate the limits

lim f(x)
x->1

and

lim f(x)
x->0^+

Answer 24

[0,infinity)

lim f(x) = e
x->1

lim f(x) = 1
x->0^+

Question 25

A continuous function y = f(x) is known to be negative atx x = 5 and positive at x = 8 Why does the equation f(x) = 0 have at least one solution between x = 5 and x = 8 ? Illustrate with a sketch.

Answer 25

f(x) = 0 has at least one solution between x = 5 andx x = 8 because fis a continuous function on the closed interval [5, 8] in any f(5) y 0 =f(c) some c in [5,8]

Question 26

Letf * (x) = |x|/x Then f(- 2) = - 1 and f(2) = 1 . Therefore, f(- 2) < 0 < f(2) , but there is no value of c between - 2 2 for which f(c) = 0 Does this fact violate the Intermediate Value Theorem? Explain.

Answer 26

It does not violate the Intermediate Value Theorem because 0 is in [- 2, 2] , but f(x) is not continuous at 0.