Calculus BC Review

Derivatives

100

f(x) = 3x²-14x³-sin(x²)

What is f'(x)?

6x-42x²-2xcos(x²)

200

f'(x)=sec(x)

Find f"(x)

f"(x)=sec(x)tan(x)

300

Find the derivative of the following: 

∫(2x-x²)dx

2x-x²

400

g(x)=10tan(x)−2cot(x)

Find the derivative

g′(x)=10sec²(x)+2csc²(x)

500

What rules of calculus are used to solve this problem? 

f(x)=6xln(x²) find f'(x)

Chain rule and product rule

Integrals

100

∫(6x⁵−18x²+7)dx

Integrate

x⁶−6x³+7x+C

200

∫sin(x)dx

Solve

-cos(x)+C

300

∫(1/1+x²)

Solve

arctan(x)+C

400

∫1/(x+5)^1/2dx

2(x+5)^1/2+C

500

Find the average value for 

f(x)=8x−3+5e^2-x on [0,2]

0.5(5+5e²)

Limits

100

True or False: a limit can still exist if the value it is approaching is not defined

True

200

What are the components of the definition of continuity?

The limit as x approaches c exists and equals f(c)

300

limx→2(x²)

4

400

limx→−5 (x²−25/x²+2x−15)

Solve

5/4

500

Use L'Hospital's Rule 

limx→2(x³−7x²+10x/x²+x−6)

-6/5

Particle Motion

100

A particle's position is defined by x(t)=x⁴-3x³+2x²-x+7

What direction is the particle moving at time t=4?

The particle is moving to the right

200

A particle's position is defined by x(t)=x⁴-3x³+2x²-x+7

Is the particle slowing down or speeding up at time t=1

The particle is speeding up

300

A particle's position is defined by x(t)=x⁴-3x³+2x²-x+7

Is the particle moving toward or away from the origin at time t=2

Away from the origin

400

dx/dt=2x² ; dy/dt=4y³ ; At time t=0 , the particle's x-coordinate is 1

Find the x-coordinate of the particle at t=2

19/3

500

What is the parametric formula for arc length?

∫((dx/dt)²+(dy/dt)²)^1/2

Random Review

100

What is the formula for the chain rule?

f'(g(x))g'(x)

200

f(x)=x²-4x

Find the relative extrema and justify your answer.

The graph of f has a relative minimum at x=2 since f'(x) changes from negative to positive at x=2.

300

f"(x)=4x³

Is f concave up or down at x=3

Concave up

400

x²+y³=4

Find dy/dx

-2x/3y²

500

What components have to be correct for mean value theorem to apply?

For a function ‍f that's differentiable over an interval from ‍a to b to, that there exists a number ‍c on that interval such that ‍f'(c) is equal to the function's average rate of change over the interval.