Limits
The limit to the following function when x approaches 1
f(x) = (x^2 - 1) / (x - 1)
What is 2?
The derivate to the following function
f(x) = -cos(x)
What is sin(x)?
The integral for the following function from 0 to pi
f(x) = sin(x)
What is 2?
d/dx (f/g)
What is (gf' - fg')/g^2?
f(x) when f'(x) and f"(x) is positive
What is increasing?
The limit to the following function when x approaches 25
f(x) = (sqrt(x) - 5) / (x - 25)
What is 1/10?
The derivate to the following function
f(x) = 3x^2(x^3 - 2x)
What is 3(5x^4 - 6x^2)?
The integral for the following function from 0 to 2pi
f(x) = cos^2(x)
What is pi?
d/dx (sin^-1(x))
What is 1/(1 + x^2)?
f(x) when f'(x) is increasing and f"(x) is positive
What is concave up?
The limit to the following function when x approaches infinity
f(x) = (cos(x) - 1) / x^2
What is 0?
The derivate to the following function
f(x) = 9(e^x)cos(x)
What is 9(e^x(cos(x)) - e^x(sin(x))?
The antiderivative for the following function
f(x) = (2x + 1) / (x + 5)^3
What is (-(2 / (x + 5)) + 9 / (2(x + 5)^2 +C)?
The three rules for a function to be continuous at a point
What are the function must be defined at that point, limit must exist at that point, and the value of the function at that point must equal the value of the limit at that point?
f(x) when f"(x) is 0 or undefined
What is have a possible point of inflection?
The limit to the following function when x approaches 3
f(x) = (x - 1/3) / (x - 3)
What is DNE?
The implicit derivate to the following function
sin(x - y) = (1 + x)
What is -(1 - cos(x - y)) / (cos(x - y))?
The antiderivative for the following function
f(x) = sin(x)cos^3(x)
What is (-1/4cos^4(x) + C)?
L'Hospital's Rule
What is if the limit of a function is indeterminate (0/0) then you can take the derivatives of the top and bottom functions separately in order to find the limit?
The tangent line compared to the true value when f"(x) is positive
What is less than the true value?
The limit to the following function when x approaches -infinity
f(x) = (x^3 - x^2 + 37) / (3x^2)
What is infinity?
The implicit derivate to the following function
f(x) = (x - y) = 1 + x/y
What is (y - y^2) / (-y^2 + x)?
The antiderivative for the following function
f(x) = e^x / (e^x + e^-x)
What is (1/2ln|e^2x + 1) + C)?
Rolle's Theorem
What is if f is differentiable for all values of x in (a, b) and f is continuous at x=a and x=b, and f(a) = f(b) = 0, then there's at least one number where x=c in (a, b) such that f'(c) = 0
The acceleration when velocity is positive and the particle is speeding up
What is positive?