Exam 2 Flashcards
For what values of a does limr * (x) =r(a) if r is a rational function? x->a
Those values of a for which the denominator of the function r is not zero.
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?
If f(x) is not continuous, then it is not a polynomial
Is the lim = infinity and lim = -infinity.
x-> 10^- x->10^+
The limit of 10 dosent exist but what exists at 10?
Vertical asymptote
Find all vertical asymptotes of the given function.
g(x) = (2x)/(x - 5)
x=5
Given a function f, what does f’ represent?
The instantaneous rate of change at any point in the domain
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 = __
(f(x)-f(a))/(x-a)
lim ((f(x)-f(a))/(x-a)
x->a
tangent
a
What is true about the graph of f(x) = |x| at the point (0, 0) ?
The graph has no tangent line.
The function f(x) = |x| has a derivative at x = 0 .
False, since the limit from the left and the limit from the right are not equal, the derivative does not exist
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?
The instantaneous change in the height with respect to the volume of water in the vase is constant
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?
f’(x)+3.2
is differentiable at a, must be continuous at a?
Yes, if is differentiable at a, then fis continuous at a
If f is continuous at a, must be differentiable at a?
No continuous at a It not necessarily true that the limit that defines f’ at a
The derivative of e to the power of any number will always be?
0
What is the maximum number of horizontal asymptotes that a function can have?
Two
True or False. A function can cross its horizontal asymptote
True
lim e^x
x->infinity
Infinity
lim e^x
x->-infinity
0
lim e^-x
x->infinity
0
lim (sin(7x))/19x
x->infinity
0
lim 3x^25
x->-infinity
Look to see if powers is even or odd
Answer= -infinity
limx^-7
x->infinity
0
Determine the end behavior
f(x) = sin(5x)
For positive and negative infinity
Both do not exist
Give the three conditions that must be satisfied by a function to be continuous at a point
lim f(x) = f(a) x->a
f(a) is defined
lim x a f(x) exists
x->a
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^+
[0,infinity)
lim f(x) = e x->1
lim f(x) = 1 x->0^+
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.
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]
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.
It does not violate the Intermediate Value Theorem because 0 is in [- 2, 2] , but f(x) is not continuous at 0.
