BC Calc AP Review

Derivatives

100

Find the derivative e^8x

8e^8x

200

Find the derivative x³sinx

3x²sinx+x³cosx

300

Find the derivative 

lntan(5x^2)

10xsec^2(5x^2)/tan(5x^2)

400

Find the derivative of 

sec^3(x^5)

15x^4sec^3(x^5)tan(x^5)

500

Find the derivative 

tan^-1(x^2)

(2x)/(1+x^4)

Integrals

100

int(x^2+5x+1)dx

x^3/3+5/2x^2+x+C

200

d/dx(int_x^(5x)lnwdw)

5ln(5x)-lnx

300

int(x^3e^(2x^4)dx)

1/8e^(2x^4)+C

400

int(x^2sinx)dx

-x^2cosx+2xsinx+2cosx+C

500

int((9x-3)/(x^2-2x-3))dx

3ln(x+1)+6ln(x-3)+C

Series

100

sum_(n=1)^(oo)(1/n^(3/4))

diverging by p-series (3/4<1)

200

Find the sum of 

sum_(n=2)^oo2^n/3^(n-1

4

300

Write the power series for 

6/(1-x^2)

sum_(n=0)^oo6(x)^(2n)

400

Find the Taylor series for sin(3x).

sum_(n=0)^oo(-1)^n(3x)^(2n+1)/((2n+1)!)

500

Find the radius of convergence of 

sum_(n=1)^oo(-1)^n(x-2)^n/3^n

3

Theorems

100

Name the theorem:

If f is continuous and f(a) = 7 and f(b)=10, there is a value c, where a<c<b where f(c) = 8.

IVT (Intermediate Value Theorem)

200

Name the theorem: If f is continuous and differentiable and a<c<b, then 

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

MVT (Mean Value Theorem)

300

If f(1) =2, f'(1)=0 and the 2nd derivative is below, is x=1 a max, min or neither? Why?

(d^2y)/(dx^2)=xy^2

A min since the 2nd derivative is positive and the graph is concave up.

400

Find the error if the first 3 terms are used to approximate the series 

sum_(n=3)^(oo)(-1)^n/(n^2)

1/36

500

Write the formula for Lagrange Error for a Taylor polynomial of degree n

f^(n+1)(z)(x-a)^(n+1)/((n+1)!)

Hodge Podge

100

Find the area for one petal of the rose 

r=5sin(6theta)

1/24int_0^(2pi)((5sin(6theta))^2d(theta)

200

Find the derivative of 

xy+y^2=8x

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

300

lim_(x->0)(e^x-x-1)/x^2

1/2  L'hopital

400

Find the second derivative if x(t)=t^2 and y(t)=sint

(-2tsint-2cost)/(8t^3)=(-tsint-cost)/(4t^3)

500

Which rule would you have to use to prove the following series is converging? 

sum_(n=1)^oosin^2(n)/(n^3)

Direct comparison to 

1/n^3