MAT120 Exam 2 Concepts

f f is continuous on ( a , b ) and f ( x ) ≠ 0 for all x in ( a , b ) , then either f ( x ) > _______ for all x in ( a , b ) or f ( x ) < ______.

0, 0

_______ and _______ asymptotes help describe the behavior of functions that are unbounded near x = a . ________ and ________ asymptotes are used to describe the behavior of functions as x assumes arbitrarily large positive values or arbitrarily large negative values.

Infinite limits, vertical Limits at infinity, horizontal

We write lim x → c f ( x ) = L or f ( x ) → L as x → c if the functional value f ( x ) is close to the single real number ______ whenever x is close, but not equal to ______

L, C

The derivative has various applications and interpretations, including For each x in the domain of f ′ , f ́ ( x ) is the ______ of the line tangent to the graph of f at the point (x, f(x)). For each x in the domain of f ′ , f ́ ( x ) is the ______ rate of change of y = f ( x ) with respect to x. If f ( x ) is the position of a moving object at time x , then v = f ′ ( x ) is the ______ of the object at that time.

Slope Instantaneous Velocity

For y = f ( x ) , the _______ from x = a to x = a + h is 𝑓 ( 𝑎 + ℎ ) − 𝑓 ( 𝑎 ) ( 𝑎 + ℎ ) − 𝑎 = 𝑓 ( 𝑎 + ℎ ) − 𝑓 ( 𝑎 ) ℎ ; ℎ ≠ 0 .

average rate of change

A function f is _____ at the point x = c if lim x → c f ( x ) exists; f ( c ) exists; lim x → c f ( x ) = f ( c )

continuous

A function is ____ on the open interval (a, b) if it is continuous at every point on the interval.

continuous

Intuitively, a function is ______ over an interval if its graph can be drawn without removing a pen from the paper.

continuous

For y = f ( x ) , we define the ______ of f at x , denoted f ′ ( x ) , by f ′ ( x ) = lim h → 0 𝑓 ( x + ℎ ) − 𝑓 ( x ) ℎ if the limit exists. If f ′ ( x ) exists for each x in the open interval ( a , b ) , then f is said to be ______ over (a,b).

derivative, differentiable

Suppose f ( x ) is a function defined in an open interval containing the number a . One of the most important limits in calculus is the limit of the _________

difference quotient

When a graph is broken, or disconnected, at a point x = c , the function is said to be _______at x=c.

discontinuous

The average rate of change is the change in the function value ________ the change in the input value.

divided by

limIf lim x → c f ( x ) = L , L ≠ 0 , and lim x → c g ( x ) = 0 , then lim f(x)/g(x) _______

does not exist

For a (two-sided) limit to exist, the limit from the left and the limit from the right must _________

exist and be equal

Lim x --C f(x) = _______for f any polynomial function or any rational function with nonzero denominator at x = c .

f(c)

If lim x → c f ( x ) = 0 and lim x → c g ( x ) = 0 , then lim x → c f ( x ) g ( x ) is said to be ________ or more specifically, a 0/0 ______

indeterminate, indeterminate form

For y = f ( x ) , the ______ at x = a is lim h → 0 𝑓 ( 𝑎 + ℎ ) − 𝑓 ( 𝑎 ) /ℎ , if this limit exists.

instantaneous rate of change

The existence of a limit at c ______ the value of the function at c.

is independent of

We write lim x → c − f ( x ) = K and call K the limit from the _______ if f ( x ) is close to K whenever x is close to, but to the ______ of c , on the real number line.

left, left

The ______ of a function at some value x = c involves examining the behavior of the output values of the function as the input values get increasingly closer to the value c .

limit

If _______ then lim x → ∞ f ( x ) = lim x → − ∞ f ( x ) = 0 , and the line y = 0 is a horizontal asymptote of f ( x ). If _______ then lim x → ∞ f ( x ) = lim x → − ∞ f ( x ) = a m b n , and the line _______ is a horizontal asymptote of f (x). If _______ then lim x → ∞ f ( x ) and lim x → − ∞ f ( x ) will be ∞ or − ∞ , depending on m , n , a m , and b n , and f ( x ) does not have a horizontal asymptote.

m < n m=n , y= a_m/b_n m>n

Let p ( x ) = a n x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 be a polynomial function with n ≥ 1 an even integer and a n a negative real number. Then lim x → ∞ p ( x ) is ______ and lim x → − ∞ p ( x ) is ______

negative, negative

Let p ( x ) = a n x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 be a polynomial function with n ≥ 1 an odd integer and a n a negative real number. Then lim x → ∞ p ( x ) is_______ and lim x → − ∞ p ( x ) is ______.

negative, positive

Polynomial functions have _____ vertical asymptotes. A vertical asymptote of a rational function can occur only at a _____ of its denominator.

no, zero

A derivative f ′ ( x ) exists whenever f ′ ( x ) = lim h → 0 𝑓 ( x + ℎ ) − 𝑓 ( x ) / ℎ exists. If the limit does not exist at x = c , we say that the function f is ______ at x=c.

nondifferentiable

A real number x is a ______ number for a function f if f is discontinuous at x or f (x) = 0

partition

Let p ( x ) = a n x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 be a polynomial function with n ≥ 1 an odd integer and a n a positive real number. Then lim x → ∞ p ( x ) is ______ and lim x → − ∞ p ( x ) is ________

positive, negative

Let p ( x ) = a n x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 be a polynomial function with n ≥ 1 an even integer and a n a positive real number. Then lim x → ∞ p ( x ) is ________ and lim x → − ∞ p ( x ) is _________

positive, positive

A function f is said to be continuous on the _____ at x = c if lim x → c + f ( x ) = f ( c ) . A function f is said to be continuous on the ______ at x = c if lim x → c − f ( x ) = f ( c ) .

right, left

A function is said to be at continuous on the closed interval [ a , b ] if it is continuous on the open interval ( a , b ) and is continuous both on the ______ at a and on the ______ at b.

right, left

We write lim x → c + f ( x ) = K and call K the limit from the ______ if f ( x ) is close to K whenever x is close to, but to the _______ of c , on the real number line.

right, right

A line containing two points on the graph of a function is called a _______

secant line

Given y = f ( x ) , the ______ of the graph at the point of ( a , f ( a ) ) is given by lim h → 0 𝑓 ( 𝑎 + ℎ ) − 𝑓 ( 𝑎 ) ℎ provided the limit exists. In this case, the ______ line to the graph is the line through ( a , f ( a ) ) with slope given by the limit.

slope, tangent line

The line x = a is a ______ asymptote for the graph of y = f ( x ) if f ( x ) → ∞ or f ( x ) → − ∞ as x → a + or x → a − .

vetical

If f ( x ) = n ( x ) d ( x ) is a rational function with d ( c ) = 0 and n ( c ) ≠ 0 , then the line _______ is a ________ asymptote of the graph of f.

x=c , vertical