Jeopardy

Derivatives

100

Find the derivative of 2tan(2x)

4sec²(2x)

200

Find the derivative of 2x²cos(x)

4xcos(x) - 2x²sin(x)

300

Find the derivative of 6x²/2e^x

(24xe^x-12x²e^x)/(2e^x)²

400

Find the Derivative of (1 + e^xsin(x))⁶

6(1 + e^xsin(x))⁵(e^xsin(x) +e^xcos(x))

500

Find the deriative of (3e^2xx⁶))/tan(x)

((6e^2xx⁶+18e^2xx⁵)tan(x) - 3e^2xx⁶sec²(x))/tan²(x)

Integrals

100

Find the indefitnite integral of (sin(x) + 2)dx

-cos(x) + 2x + C

200

Find the definite integral from x = 0 to x = pi/2 of x²sin(x)dx

pi - 2

300

Find the definite integral from x = 4 to x =5 of (4x+6)/(x²-9)dx

2.213(calc) 6.104(by hand)

400

Find the integral from 1 to infinity of xe^-xdx

2/e

500

Find the definite integral from x=0 to x=pi/4 of (2xe^2x+sec²(x))dx

(pi*e^pi/2)/4 - (e^pi/2)/2 + 0.5

Miscellaneous

100

What is the first, second, and third derivative of position?

Velocity, acceleration, jerk

200

Find the linearization of f(x) = sqrt(1+x) at x=0, and use it to approximate sqrt(1.02) without a calculator.

L(x) = 1 + x/2
L(0.02) = 1.01

300

What is the Mean Value Theorem?

If  a function f(x) is continous on the interval [a,b] & differentiable on the interval (a,b), then there exists at least one number on the interval (a,b) where the instantenous rate of change = average rate of change

400

What are the conditions for the integral test?

Function must be decreasing, positive, and continuous 

500

Find the average value from x=1 to x=5 of the integral (3x²-2e-^2x)dx

About 31.034

Sequences and Series

100

What is the Lagrange Error Bound Equation?

|R_n(x)| < (|x-c|ⁿ⁺¹)/(n+1)! *max|fⁿ⁺¹(z)|

200

If the series E_n=1(-1)ⁿ⁺¹/(2n+1) is appromximated by the partial sum with 15 terms, what is the alternating series error bound?

1/33

300

Find the second degree Taylor Polynomial for f(x) = e^2x+3cos(x) centered at x=0

P₂(x)= 4 + 2x + x²/2

400

Let h(x) = 12/(3+x). Find a power series for h centered at x = 0.

h(x) = E_n=0 4(-x/3)ⁿ

500

Find the interval of convergence of E_n=0 3((x-1)/5)ⁿ

(-4, 6)

Vectors

100

Find all the points of inflection on the cure x = t²-5 and y = 2sin(t)

(2.829, 0.674)

200

If r = 1 - cos(0), find the area of the curve from 0= pi/2 to 0 = 3pi/2 

0= theta

4.356 units²

300

Find the length of the curve x = cos(t), y = sin(t) for 0<t<2pi

2pi

400

If v(t) = <3t²-2t, 1+cos(tpi), find the position of the particle at time = 3,with an original position of <2,6>.

<20,9>

500

Find the area of the region that lies inside the circle r=1 and outside the cardioid r = 1-cos(0) from 0 to pi/2.

1.215