Calculus
State the Intermediate Value Theorem
If the function f(x) is continuous on [a,b], and y is a number between f(a) and f(b), then there exists at least one number x = c in the open interval (a,b) such that f(c) = y.
Symbolically write the Quotient Rule.
[(f'(x) * g(x)) - ((f(x) * g'(x))] / (g(x))^2
Take the derivative of sin(x).
cos(x)
Take the derivative of e^x.
e^x
Take the derivative of sec(x).
sec(x)tan(x)
Take the derivative of tan(x).
sec^2(x)
Take the derivative of arctan(x).
u' / 1+u^2
Take the derivative of arcsec(x).
u' / |u| √(u^2 - 1)
State the Fundamental Theorem of Calculus.
∫ from a - b of f(x) dx = F(b) - F(a) where F'(x) = f(x)
What is sin 30⁰?
(1/2)
Take the derivative of csc(x).
-csc(x)cot(x)
Take the derivative of cot(x).
-csc^2(x)
Take the derivative of cos(x).
-sin(x)
Take the derivative of arccot(x).
-u' / 1 + u^2
Take the derivative of arccsc(x).
-u' / |u| √(u^2 - 1)
Take the derivative of arccos(x).
-u' / √(1 - u^2)
What is the integral of ln(x)?
...
What is sin(0)?
0
What is tan(π)?
0
What is 30 degrees in radians?
0.524 R
Take the derivative of log∨a(x).
1 / ln(a) * x
Take the derivative of ln(x).
1/x
What does an inflection point signify?
A change in concavity
What is the Mean Value Theorem?
If the function f(x) is continuous on [a,b], and it is also differentiable on the open set (a,b), you can always find an x = c in (a,b) such that f'(c) = [f(b) - f(a)] / (b-a)
Does the second derivative need to exist for there to be an inflection point?
No, the second derivative could be undefined
How do you find critical points?
Set f' equal to 0
How do you find a local maximum?
Set the derivative and set it equal to zero (or undefined) and check it.
How do you find a local minimum?
Set the derivative and set it equal to zero (or undefined) and check it.
What is the Product Rule?
[f(x) * g'(x)] + [f('x) * g(x)]
How do you find the average velocity?
final position - initial position / total time
Take the derivative of a^x.
ln(a) * a^x
State the Chain Rule for Differentiation
n(f(x))^n-1 * f'(x)
Take the derivative of arcsin(x).
u' / √1-u^2)