Limits and Continuity
This mathematical concept describes the behavior of a function as the input approaches a specific value (or +- infinity).
What is a Derivative?
Find integral of 5x^4.
What is x^5?
The geometric interpretation of a derivative at a point
What is a tangent line?
This theorem links differentiation and integration together.
What is FTC?
What is 20x^4 + 6x?
Evaluate: lim [x->2] ([x^2-4]/[x-2])?
What is 4?
Find the antiderivative of 2x*e^x^2.
What is e^x^2 + C?
A box with a square base must have a volume of 32cm^3. What are the dimensions for the most minimized surface area?
What is 2sqrt(2) x 2sqrt(2) x 4?
This theorem states that for a continuous function on [a,b], there exists c where f'(c) = (f(b)-f(a))/(b-a)
What is MVT?
differentiate f(x)=(x^2)⋅sin(x).
What is 2xsin(x)+x^2cos(x)?
This rule resolves limits that evaluates to 0/0 or inf/inf.
What is L'Hopital's Rule?
Calculate the integral of 3x^2-2x+1 from 0 to 4.
What is 48?
A ladder slides down a wall at 2ft/s. Find how fast the base moves when the top is 6ft high (Ladder length is 10ft.)
What is 3/2 ft/s?
This guy invented calculus around the same time with Newton, but is often overlooked.
Who is Leibniz?
find d/dx of sqrt(3x^2 + 5)
What is 3x/(sqrt(3^2+5)
Evaluate: lim [x->7] ([y^2 - 4y -21]/[3y^2 - 17y - 28])
What is 2/5?
Find the integral of xcos(x) using IBP.
What is xsin(x) + cos(x) + C?
Calculate the area between y=x^2 and y=2x−x^2.
What is 1/3?
This theorem guarantees a function’s value between two points if it’s continuous.
What is Intermediate Value Theorem?
find dy/dx of x^3+y^3 = 6xy
What is (2y-x^2)/(y^2-2x)?
A function with f(x) = 0 when x = -2, a vertical asymptote at x=1 and a removable discontinuity at x=3.
What is f(x) = ([x-3][x+2])/([x-3][x-1])
Find the volume of the solid formed by rotating y=x around the x-axis from x=0 to x=4.
What is 8pi?
A particle's acceleration is 6t. If v(0) = 4 and x(0) = 10, what is x(2)?
What is 26?
This theorem states that if a function f is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists a number c in the open interval (a, b) such that f'(c) = 0. (Hint: A special case of MVT)
What is Rolle's Theorem?
find f''(x) of ln(x^2+1)
What is 2(1-x^2)/(x^2+1)^2