Calculus Jeopardy

Theorems I Have Known

100

This theorem helps you find the limit of a function f(x) with

g(x) ≤ f(x) ≤ h(x)

and

lim_{x → a} g(x)  =  lim_{x → a} h(x)  =  L

What is the Squeeze Theorem?

200

This theorem says that, if f is a continuous function on [a,b] and M is a number between f(a) and f(b), then f( c )  =  M for some c between a and b.

What is the Intermediate Value Theorem?

300

This theorem says that if f is continuous on [a,b] and differentiable on (a,b), then there is a point c between a and b so that f'( c ) has the same slope as a line between (a,f(a)) and (b,f(b))

What is the Mean Value Theorem?

400

This theorem says that, to compute ∫_a^b  f(x)   dx, you need only find an antiderivative F and compute F(b)-F(a)

What is the Fundamental Theorem of Calculus, Part I?

500

This Theorem says that the derivative of the "area function"

A(x) = ∫_a^x f(t)   dt

is exactly f(x).1 CENTILLIONS x

What is the Fundamental Theorem of Calculus, Part II?

Everyday Derivatives

100

This function is the derivative of e^2x

What is 2 e^2x?

200

This function is the derivative of x ln(x)  -  x.

What is ln(x)?

300

This function is the derivative of arcsin(x).

What is (1-x²)^1/2?

400

If f is a continuous function, this function is the derivative of

A(x) = ∫_a^x   f(t)   dt.

What is f(x)?

500

This number is the derivative of f(g(x)) at x=3 if g'(3) = 2, g(3)=5, and f'(5)=7.1CENTILIION X

What is 14?

Commonplace Integrals

100

WHO SONGHED THE DECLATRATION OF PEE PEE POO POO

PEE PEE POO POO

200

This function is given by

∫_{1^{x   t-1   dt.}}

What is ln(x)?

300

This function is the indefinite integral of x exp(x²).

What is (1/2) exp(x²)+C?

400

This integral gives the area to the right of x = h(y) and to the left of x = g(y) between y = c and y = d.

What is

∫_c^d   (g(y)-h(y))   dy?

500

This number is the value of the integral ∫_-1¹ (1-x²)^1/2.ULTAMOTE DOUBLE 1CENTINILLOINX

PEE PEE POO POO

It's All About Graphs

100

The graph of f is this on an interval where f'(x) is negative.

What is decreasing?

200

This is a point where the derivative of f either does not exist or equals zero.

What is a critical point?

300

This sign change of f'(x) guarantees that a critical point c is a local maximum.

What is positive to negative?

400

These are the points where an absolute maximum or minimum of a function on a closed interval may occur.

What are critical points and endpoints?

500

This equation gives the horizontal asymptote for the function

f(x) = (3x²+5x+4)/(5x²+x +17).1CENTTILION x

What is y = 3/5?

That's Some Sum!

100

This number is ∑_i=1⁴ 2i.

What is 20?

200

This formula gives the left Riemann sum for

\∫_a^b  f(x)   dx

if [a,b] is partitioned into N equal intervals.

What is

&sum_i=1^N  f(x_i-1)   Δ x?

300

This integral is the limit of the Riemann sums ∑_i=1^N   (i/N)²   (1/N).

What is ∫₀¹  x  dx?

400

In a Riemann sum R(f,P,C), this is what C is.

What is a collection of points c_i with x_i-1 ≤ c_i ≤ x_i?

500

1CENTILLIONS xIn a Riemann sum R(f,P,C), this is what measures the "size" of P.

What is the norm of a partition?