Calculus Jeopardy By (Andrew, Bryana, and Alejandro)

Derivatives

100

(f g)’ = f‘g + fg’ exists under what rule?

= Product Rule

200

What is the formula for the power rule?

= (d/dx)(xⁿ) = nx^n-1

300

 What does (d/dx)(tan x) equal?

= sec² x

400

What does (d/dx)(e^x) equal?

= e^x

500

Differentiate: y = sec² x + tan² x

= 4sec² x tan x

Integrals

100

Find the integrals of Sf(u) du?

= F(u) + C

200

Find the integral of _aS^b c dx?

= c(b-a)

300

1. Find the integral of Scos u du?

= sin u + c

400

Find the integral of Sk dx?

= kx + c

500

Find the integral of S x^(1/3) (x - 4) dx

=  (3/7) x^4/3(x-7) + C

Limits

100

Find the Limit: lim_x-2 x² 

= lim_x-2 2²

= 4

200

Find the Limit: lim_x-2 (4x² + 3)

= lim_x-2 4x² + lim_x-2 3

= 4(lim_x-2 x²) + lim_x-2 3

= 4(2²) + 3

= 19

300

Find the Limit: lim_x-1 (x² + x + 2) / (x + 1)

= (1² + 1 + 2) / (1 + 1)

= 4/2

= 2

400

Find the Limit: lim_x-3.14 (x cos x)

= (lim_x-3.14 x) (lim_x-3.14 cos x)

= 3.14 cos 3.14

= -3.14

500

Find the Limit: lim_x-1 (x³ - 1)/(x - 1)

= lim_x-1 ((x - 1)(x²+x+1))/(x-1)

= lim_x-1 (x²+x+1)

= 1² + 1 + 1

= 3

Theorems and Methods

100

What is the formula for the mean value theorem?

f'(c) = (f(b)-f(a))/(b-a)

200

What is one of the conditions of the Mean Value Theorem?

1. It is continuous in the closed interval [a,b]

2. It is derivable in the open interval (a,b)

300

Let f(x) = (x + 9)^1/2 and let c be the number that satisfies the Mean Value Theorem for f on the interval [0,16].

Answer: 7

400

Let f(x) = x³ + 12x² + 36x and let c be the number that satisfies the Mean Value Theorem for f on the interval -8 ≤ x ≤ -2.

Answer: -6

500

What is one of the conditions of Rolle’s Theorem?

1. It is continuous in the closed interval [a,b]

2. It is derivable in the open interval (a,b)

3. f(a) = f(b)

Area

100

Find the area of a region between intersecting curves of the lines

g(x)= x 

and

f(x)=2-x2

2-x² =x

-x²-x+2=0

-(x-2)(x+1)=0

   x=-2 or 1

Then

A =_-2∫¹ {[(2-x²) - (x)] dx = [-(x³/3)-(x²/2)+2x]¹_-2

A = 9/2

200

Find the area of a region between intersecting curves of the lines

f(x) = x²

and

g(x)= x³

Answer: ₀∫¹[(x²)-(x³)]

300

Find the area of the region bounded by the two graphs 

f(x)= x+1

and

g(x) = (x-1)²

X+1 = (x-1)²       (x=0, 3) 

∫₀³((x-1)-(x-1)²))dx

[(-x³/3)+(3x²/2)]₀³

-(3³)/3 + 3(3²)/2 -0 

=9/2

400

Use the Disk or Washer Method to solve

y=x, y=3, x=0  Revolved at the line (y=4)

Answer= 18(3.14)

500

Use the Disk or Washer Method to solve

y=3(2-x), y=0, x=0 (revolved around the y-axis)

Answer= 8(3.14)