definitions

Question 1

Intermediate Value Theorem (IVT)

Answer 1

If f continuous on [a,b], the f takes on all values between f(a) and f(b)

Question 2

Average Rate of Change on the Interval [a,b]

Answer 2

AROC = f(b)-f(a) / b-a

Question 3

Definition of a Vertical Asymptote

Answer 3

x = a is a VA of f if lim x->a f(x) = +/- ∞

Question 4

Definition of a Horizontal Asymptote

Answer 4

y = b is a HA of f if lim x->+/- ∞ f(x) = b

Question 5

Squeeze Theorem

Answer 5

Let g(x) ≤ f(x) ≤ h(x) for x ≠ a on some interval about x = a. If lim x->a g(x) = lim x->a h(x) = L, then lim x->a f(x) = L

Question 6

Definition of Continuity at a Point

Answer 6

If lim x->a f(x) = f(a), the f is continuous at x = a

Question 7

Existence of the Limit of a function

Answer 7

lim x->a f(x) = L if and only if lim x->a- f(x) = lim x->a+ f(x) = L

Question 8

Limit Definition of the Derivative

Answer 8

f’(x) = lim h->0 f(x+h)-f(x) / h

Question 9

Alternate Definition of the Derivative

Answer 9

f’(x) = lim x->a f(x) - f(a) / x - a

Question 10

Mean Value Theorem of Derivative

Answer 10

If f continuous on [a,b] and differentiable on (a,b) then there exists a c in (a,b) at which f’(c) = f(b) - f(a) / b - a

Question 11

Extreme Value Theorem

Answer 11

If f is continuous on [a,b] then f has both a min and max on [a,b]

Question 12

Interior Extremum Theorem

Answer 12

If f is differentiable on (a,b) and f has a local max/min on c (a,b), the f’(c) = 0

Question 13

Monotonic Function

Answer 13

A function is monotonic on an interval if it is always increasing or decreasing on the interval

Question 14

Concavity Definition

Answer 14

f is concave up where f’ is increasing and f” > 0

f is concave down where f’ is decreasing and f”<0

Question 15

Inflection Point Definition

Answer 15

A point in the domain of f where there is a tangent line and concanvity changes

Question 16

Critical Point definition

Answer 16

A point at which f’ = 0 or f’ DNE

Question 17

1st Derivative Test

Answer 17

If f’ changes + to - at x = c, then f has a local max at x = c
If f’ changes - to + at x = c, then f has a local min at x = c
If f’ does not change sign, there is neither a min nor a max

Question 18

2nd Derivative Test

Answer 18

If f’ = 0 and f” > 0 at x = c, then f has a local min at x = c
If f’ = 0 and f” < 0 at x = c, then f has a local max at x = c

Question 19

Points at which f is not differentiable (4)

Answer 19

Corner, Cusp, Jump, Infinite discontinuity

Question 20

Stationary Point

Answer 20

A point at which f’(x) = 0

Question 21

L’Hospital’s Rule

Answer 21

If lim x->a f(x) / g(x) results in 0 / 0 or ∞ / ∞ then lim x->a f(x) / g(x) = lim x->a f’(x) / g’(x)

Question 22

Fundamental Theorem of Calculus Parts 1 and 2

Answer 22

the integral from a to b of f(x) dx = F(b) - F(a)

the derivative of the integral from a to x of f(t) dt = f(x)

Question 23

Mean Value Theorem for Integrals

Answer 23

If f is continuous on [a,b] then at some point c in [a,b], f(c) = 1 / b-a of the integral a to b of f(x) dx

Question 24

Average Value of a Function

Answer 24

1 / b - a from the integral of a to b of f(x) dx

Question 25

Trapezoidal Rule

Answer 25

integral from a to b of f(x) dx ≈ h / 2 ( f(a) + 2 f(x₁) + 2 f(x₂) + … + 2 f(xₙ) + f(b) `)

Question 26

Volume

Answer 26

π integral a to b of ( f(x) )² dx

π integral a to b of [ ( f(x) )² - ( g(x) )² ] dx