Calculus AB Jeopardy

Limits and Continuity

100

When is a function differentiable?

The function is countinous over the interval and the left and right hand limit of f(x) equal eachother and the limit of f(x) = f(x).

200

What is the limit definition of a derivative 

f’(x)=lim_h->0 [f(x+h)-f(x)]/h

300

Evaluate lim_x→2(8−3x+12x²), if it exists.

What is the lim_x→2 = 50

400

Determine the limit in simplest form. (x²-x)/(x²-1)

What is (x)/(x+1)

500

Evaluate the limit: lim_h→0 [(6+h)² -36]/h

What is lim_h→0=12

Derivatives

100

Would you use the chain rule, power rule, quotient rule, or the product rule to find the derivative of 12x⁶+7x³+12? What is the derivative of the function?

What is the Power Rule and 72x⁵+21x²?

200

Given the function y=3(-4x-5)⁶, find dy/dx in any form.

What is dy/dx=18(-4x-5)⁵(-4)

300

Let f and g be differentiable functions where g(x)=f^-1(x)for all x. f(3)=-2, f(-2)=4, f'(3)=5, and f'(-2)=1. Find g'(-2)

What is 1/5

400

Given the function y=3√sin(5-4x), find dydx

What is -6cos(5-4x)/√sin(5-4x)

500

Given

Find dy/dx  5x²+2y²=4, find dy/dx in terms of x and y.

What is 

What is dy/dx=-5x/3y² 

Integrals

100

Solve the integral.

∫x dx

What is x²/2 + C?

200

Solve to find the integral of the equation.

 f(x)=(x² + 3)

What is x³/3 + 3x + C?

300

 Solve the integral.

∫sin(x) dx

-cos(x)+C

400

Solve the integral.

∫(x + cos(x)) dx

What is x²/2 + sin(x) + C

500

Solve for f(x) if f’(x)=g(x).

g(x)=(2x + 1) / ((x + 1)² )

What is f(x)= 2ln|x+1|+1/(x+1)+C

Theorems

100

This guarantees the existence of a point c within the interval where the derivative of the function is equal to the average rate of change over the interval. 

What is the Mean Value Theorem (MVT)

200

A function is continuous on a closed interval [a, b], and there's a value L between f(a) and f(b), then there must be a point c within the interval where the function's value equals L.

What is the Intermediate Value Theorem (IVT)?

300

a real-valued function f is continuous on the closed and bounded interval [a,b], then f must attain a maximum and a minimum, each at least once.

What is the Extreme Value Theorem (EVT)?

400

Two Parts:

Part One: If you're integrating a function f(x) from a to x, and then you differentiate the result with respect to x, you get f(x) back. In other words, the derivative of the accumulation function (the result of an indefinite integral) is the original function.

Part Two: 

The definite integral of a function f(x) from a to b is equal to the difference in the antiderivative F(x) of f(x) evaluated at b and a. So, if you can find an antiderivative F(x) of f(x), then the definite integral can be easily calculated as F(b) - F(a). 

What is the Fundamental Theorem of Calculus (FTC)?

500

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 x=c in (a, b) such that f'(c) = 0

What is Rolle’s Theorem?

Graphs

100

What is the slope of the tangent line of f(x) equal to?

What is f ‘(x)?

200

Where in f(x)’s line is f’(x) increasing and positive?

What is when it is concave up and increasing?

300

When would a graph f(x) experience a relative maxima/minima based on the graph of f’(x)?

What is when f’(x) changes signs (negative to positive vise versa)

400

What Riemann sum is the most accurate?

What is the trapezoidal sum?

500

To find the area between two curves, y=f(x) and y=g(x), on the interval [a,b] where g is greater than f.

What is ∫^b_a[g(x)−f(x)]dx?