Calculus Jeopardy

Theorems

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

 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.

 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))

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)

 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).

Fundamental Theorem of Calculus, Part II

Derivatives

100

What is the derivative of e^2x

2 e^2x

200

What is the derivative of sec(x).

sec(x)*tan(x)

300

What is the derivative of x ln(x)  -  x.

ln(x)

400

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

A(x) = ∫_a^x   f(t)   dt.

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.

14

Integrals

100

What is the antiderivative of (1+x²)^-1.

 tan(x)

200

Evaluate

∫_{1^{x   t-1   dt.}}

 ln|t|

300

What is the indefinite integral of x exp(x²).

Hint: Use substitution method

 (1/2) exp(x²)+C?

400

What 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.

∫_c^d   (g(y)-h(y))   dy

500

What is the value of the integral ∫₀¹ (x²)^1/2.

1/2?

It's All About Graphs

100

What is happening to Tte graph of f on an interval where f'(x) is negative.

decreasing

200

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

 critical point

300

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

from positive to negative

400

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

critical points and/or endpoints

500

What is the horizontal asymptote for the function

f(x) = (3x²+5x+4)/(5x²+x +17).

y = 3/5?

That's Some Sum!

100

What ∑_i=1⁴ 2.

8

200

What is ∑_i=1^N (i).

N*(N+1)/2

300

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

partition

400

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

sample points: a collection of points c_i with x_i-1 ≤ c_i ≤ x_i

500

In a Riemann sum R(f,P,C),  what measures the "size" of P?

 norm of a partition ||P||