Theorems
Fundamental Theorem of Calculus states that:
(int_{a}/{b} f(x) dx) =
(The integral from a to b of f(x), a and b being real numbers)
F(b) - F(a)
What is the derivative of:
x^4?
4x^3
What is the integral of:
12?
12x + c
If the second derivative is positive, is the equation an underestimate or an overestimate?
Underestimate
Mean Value Theorem
If (f) is continuous on ([a, b]) and differentiable on ((a, b)), there exists a point (c) in ((a, b)) such that (f'(c)) =
{f(b) - f(a)}/{b - a})
What is the derivative of:
-x^4 + 4x^2 - 5
-4x^3 + 8x
What is the integral of:
sin(x)?
-cos(x) + c
What is the only way to have removable discontinuity?
Hole
If \(f\) is continuous on a closed interval \([a, b]\), then \(f\) must have both an absolute ____ and an absolute ____ on that interval.
Maximum, Minimum
What is the derivative of:
(g(x))x(f(x))?
(g'(x))x(f(x)) + ((f'(x))x(g(x))
What is the integral of:
x^7 - 4x^3 -2
1/8x^8 - x^4 - 2x + c
How do you find the VA?
Set denominator equal to 0, solve for x.
If (f) is continuous on ([a, b]), differentiable on ((a, b)), and (f(a) = f(b)), then there is at least one (c) in ((a, b)) where (f'(c)) =
0
What is the derivative of:
ln(ln(x))?
1/xln(x)
What is the integral of:
1/2x^1/2?
x^1/2 + c
The temperature of a room, in degrees Fahrenheit, is modeled by H, a differentiable function of the number of minutes after the thermostat is adjusted. Of the following, which is the best interpretation of H'(5)=2
A) The temperature of the room is 2 degrees Fahrenheit, 5 minutes after the thermostat is adjusted.
(B) The temperature of the room increases by 2 degrees Fahrenheit during the first 5 minutes after the thermostat is adjusted.
(C) The temperature of the room is increasing at a constant rate of 2/5 degree Fahrenheit per minute.
(D) The temperature of the room is increasing at a rate of 2 degrees Fahrenheit per minute, 5 minutes after the thermostat is adjusted.
D
Intermediate value theorem:
If (f) is continuous on ([a, b]), then (f) takes every value (k) between (f(a)) and (f(b)) at some point __ in the interval.
C
What is the second derivative of:
(u)^1/2
-1/4(u)^3/2
What is the integral of:
tan(x)?
ln(sec(x)) + C
A function f is continuous on the closed interval [2,5] with f(2)=17 and f(5)=17. Which of the following additional conditions guarantees that there is a number c in the open interval [2,5] such that f'(c)=0
A) No additional conditions
B) f has a relative extremum on the interval [2,5]
C) f is differentiable on the interval [2,5]
D) The integral from 2~5 of f(x) exists
C