Calculus Jeopardy

Limits

100

lim_x->0 1/x

DNE

200

lim_x->3- (x^2-2)/(x-3)

-infinity

300

The three limit rules for continuity.

1. The limit must exist

2. f(a) must exist

3. f(a) must equal the limit

400

The limit definition of a derivative.

lim_h->0 (f(x+h)-f(x))/h 

or lim_x->a (f(x)-f(a))/(x-a)

500

lim_x->4 (x^2-3x-4)/(x-4)

5

Derivatives

100

d/dx (x(x-7)+6)

2x-7

200

d/dx (tan(x)+sec(x))^2

2(tan(x)+sec(x))*(sec^2(x)+tan(x)sec(x))

300

d/dx (4arcsin(3x))

12/((3x)^2+1)

400

Find dy/dx of 3xy+4y^2=2x

dy/dx=(2-3y)/(3x+8y)

500

A circle's radius is increasing at a rate of 4 m/s (A=pir^2). At what rate is the area increasing when it has a radius of 6m.

dA/dt=48pi

Integrals

100

∫4x dx

2x^2 +C

200

∫(5x^2+6x-7)dx |0, 4

126 2/3

300

∫4e^(4x)

e^(4x)+C

400

∫((cos(ln(x)))/x) dx

sin(ln(x))+C

500

∫(4x(ln(2x))) dx

2x^2ln(2x)-x^2+C

Theorems

100

What is the condition to use IVT?

Continuity

200

What are the conditions to use MVT?

Continuity and Differentiability 

300

Can you use IVT on the function f(x)=4x^2+6x^3+2x-7?

Yes, polynomials are continuous.

400

On f(x)=4x^2-7 [-1,6], does k=4?

Yes, -3<4<137

500

When does f(x)=2x^3-4x+8 satisfy MVT on [-2, 2].

x= ±√(4/3)

Applications of Integrals

100

Find the area between y=-4x^2+6 and the x and y axis.

4.667

200

Find the area between y=-4/(x^2-7) and y=2x in the first quadrant. 

3.640

300

Find the volume of y=-x^2+6x+2 in the first quadrant using semi-circle cross-sections.

167.931

400

Find the volume of the area between x=4 and y=3x^2 and the x-axis rotated around the x-axis

1843.2 pi

500

Find the area between y=5x^2+6x+2, y=34, and the y-axis in the first quadrant rotated around the x-axis.

922.667 pi