Chapter 3

Question 1

What is a “critical number” of a function?

Answer 1

A number in the domain of the function where the derivative of the function at that point is either 0 or undefined

Question 2

What is the Extreme Value Theorem?

Answer 2

If f(x) is a continuous function on N = [a, b], then f(x) will attain an absolute maximum and absolute minimum on N

Question 3

What is Rolle’s Theorem?

Answer 3

If a function f(x) is continuous on [A, B] and differentiable on (A, B), and f(A) = f(B), then it is always possible to find a number C where A<C<B and f’(C) = 0.

Question 4

What is the Mean Value Theorem?

Answer 4

If a function f(x) is continuous on [A, B] and differentiable on (A, B), then it is always possible to find a number C where A<C<B and f’(C) = [ f(B) - f(A) ]/[ B - A ].

Question 5

Suppose f(x) is a function that is continuous on [A, B] and differentiable on (A, B). What must also be true so that a number C where A<C<B and f’(C) = 0 can always be found?

Answer 5

f(A) = f(B)

Question 6

What is the formula for the linearization of function f(x) at x = a?

Answer 6

L(x) = f(a) + f’(a) * (x - a)

Question 7

How is a local minimum N of a function f(x) defined?

Answer 7

Any value of N where f’(N) = 0, f’(N-h) < 0, and f’(N+h) > 0, where h is arbitrarily small but greater than zero
Any value of N where f’(N) = 0 and f’‘(N) > 0
Any value of N where f’(N) = 0 and f’ flips from negative to positive when crossing N

Question 8

How is a local maximum M of a function f(x) defined?

Answer 8

Any value of M where f’(M) = 0, f’(M-h) > 0, and f’(M+h) < 0, where h is arbitrarily small but greater than zero
Any value of M where f’(M) = 0 and f’‘(M) < 0
Any value of M where f’(M) = 0 and f’ flips from positive to negative when crossing M

Question 9

Let c be a critical number of the function f(x), and let h be arbitrarily small but greater than zero. If f’(c-h) > 0 [positive] and f’(c+h) < 0 [negative], is c a local minimum or a local maximum?

Answer 9

c is a local maximum of f(x)

Question 10

Let c be a critical number of the function f(x), and let h be arbitrarily small but greater than zero. If f’(c-h) < 0 [negative] and f’(c+h) > 0 [positive], is c a local minimum or a local maximum?

Answer 10

c is a local minimum of f(x)

Question 11

Let c be a critical number of the function f(x), and let h be arbitrarily small but greater than zero. If f’(c-h) > 0 [positive] and f’(c+h) > 0 [positive], is c a local minimum or a local maximum?

Answer 11

c is not a local minimum or local maximum of f(x)

Question 12

Let c be a critical number of the function f(x), and let h be arbitrarily small but greater than zero. If f’(c-h) < 0 [negative] and f’(c+h) < 0 [negative], is c a local minimum or a local maximum?

Answer 12

c is not a local maximum or local minimum of f(x)

Question 13

How do you determine on what intervals a function f(x) is increasing or decreasing?

Answer 13

1. Find all critical values of f(x) [where f’(x) = 0, we will call this set C]
2. Determine relevant intervals [their endpoints will either be -∞, +∞, or an element of C]
3. Evaluate f’(x) for one input in each interval [negative = f(x) decreases on the interval, positive = f(x) increases on the interval]

Question 14

Why is f’(x) used to determine if f(x) is increasing or decreasing on an interval?

Answer 14

Because f’(x) is defined as the instantaneous rate of change [increase or decrease] of f(x)

Question 15

How is an inflection point P of a function f(x) defined?

Answer 15

Any value of P where f’‘(P) = 0 and [ f’‘(P-h) ]/[ f’‘(P+h) ] < 0, where h is arbitrarily small but greater than zero
Any value of P where f’‘(P) = 0 and the sign of f’’ flips when crossing P

Question 16

How do you determine on what intervals a function f(x) is concave up or concave down?

Answer 16

1. Find all critical values of f’(x) [where f’‘(x) = 0, we will call this set C]
2. Determine relevant intervals [their endpoints will either be -∞, +∞, or an element of C]
3. Evaluate f’‘(x) for one input in each interval [negative = f(x) is concave down on the interval, positive = f(x) is concave up on the interval]

Question 17

Why is f’‘(x) used to determine if f(x) is concave up or concave down on an interval?

Answer 17

Because f’‘(x) is defined as the instantaneous rate of change [increase or decrease] of f’(x), and a function is concave up [curving upward] when its instantaneous rate of change is increasing and concave down [curving downward] when its instantaneous rate of change is decreasing.

Question 18

What does it mean for a function to be concave upwards on an interval?

Answer 18

The function’s instantaneous rate of change increases over that interval
The function lies above all lines tangent to itself over that interval
The function appears to curve upward, as if cupping something above itself

Question 19

What does it mean for a function to be concave downwards on an interval?

Answer 19

The function’s instantaneous rate of change decreases over that interval
The function lies below all lines tangent to itself over the interval
The function appears to curve downward, as if cupping something below itself

Question 20

What is a vertical asymptote?

Answer 20

A vertical line x=a where lim{x→a+}(f(x)) = ±∞ and lim{x→a-}(f(x)) = ±∞

Question 21

What is a horizontal asymptote?

Answer 21

A horizontal line y=a where lim{x→-∞}(f(x)) = L and lim{x→+∞}(f(x)) = L

Question 22

What is a slant asymptote?

Answer 22

A line y=mx+b that f(x) approaches but never touches as x→±∞

Question 23

Given a rational function f(x) with a numerator with leading term ax^P and a denominator with leading term bx^Q, what can be said about the function’s asymptotes?

Answer 23

If P<Q, then f(x) has a horizontal asymptote at y = 0
If P=Q, then f(x) has a horizontal asymptote at a/b
If P>Q, then f(x) has no horizontal asymptotes, but a slant asymptote may exist if P=Q+1

Question 24

Given a rational function f(x) with a numerator with leading coefficient ax^P and a denominator with leading coefficient bx^Q where P<Q, what can be said about the function’s asymptotes?

Answer 24

f(x) has a horizontal asymptote at y = 0

Question 25

Given a rational function f(x) with a numerator with leading coefficient ax^P and a denominator with leading coefficient bx^Q where P=Q, what can be said about the function's asymptotes?

Answer 25

f(x) has a horizontal asymptote at a/b

Question 26

Given a rational function f(x) with a numerator with leading coefficient ax^P and a denominator with leading coefficient bx^Q where P

Answer 26

If P

Question 27

How can you find a slant asymptote for any rational function that has one?

Answer 27

Use long division to divide the numerator by the denominator; the equation of the slant asymptote is the quotient (ignore the remainder once the quotient is a linear function)

Question 28

What is the "1st Derivative Test"?

Answer 28

Let c be a critical number of a continuous function f. 1) If f'(x) changes from + to - as x passes c, then f has a local maximum at c. 2) If f'(x) changes from - to + as x passes c, then f has a local minimum at c. 3) If the sign of f'(x) does not change as x passes c, then there is no local minimum or local maximum at c.

Question 29

What is the "2nd Derivative Test"?

Answer 29

Let f''(x) be continuous near c. 1) If f'(c) = 0 and f''(c) > 0, then f has a local minimum at c. 2) If f'(c) = 0 and f''(c) < 0, then f has a local maximum at c.

Question 30

What is the general strategy for maximizing/minimizing the value of a variable F given its relation to other variables?

Answer 30

1. Rewrite F in terms of one variable [e.g., F(X)] 2. Find F'(X) and the critical points of F(X) 3. Use the critical points and constraints of X to find the max/min values of F(X) 4. Solve F(X) for the desired value

Question 31

What is Newton's Method?

Answer 31

A repeated process used to approximate the root(s) of a function when given the form f(x) = 0

Question 32

How do you find term k+1 in a Newton's Method sequence?

Answer 32

x{k+1} = x{k} - f( x{k} )/f'( x{k} )

Question 33

Where on f(x) will Newton's Method fail?

Answer 33

Where f'(x)=0 [critical numbers]

Question 34

What is the antiderivative of a function f(x)?

Answer 34

Any function F(x) that satisfies F'(x) = f(x)