Calculus Review

Derivative

100

If f(x) = 7x - 3 + lnx, then f'(1) =

8

200

If y = xsinx, then dy/dx =

sinx + xcosx

300

If y = (x³ - cosx)⁵, then y' =

y'= 5(x³- cosx)⁴ * (3x² + sinx)

400

If f(x) = ln(x + 2 + e^-4x), then f'(0) =

-1

500

If f(x) = e^(1/x), then f'(x) =

(-1/x²) * e^(1/x)

Integrals

100

∫secxtanxdx =

secx + c

200

The average value of the function g(x) = 2√(1+5x) on the interval [-4,0] is:

6.349

300

∫₂⁵ f(t)dt = 9, what is: ∫₂⁵ (3 * f(t) - 4) dt?

15

400

Find the total distance tranveled in the first five seconds, for a particle whose velocity is given by: 

v(t) = 8e^-3t - √t

8.692

500

A particle moves along a line so that its acceleration for t ≥ 0 is given by a(t) = (t +3)/(√(t³ + 1)). If the particle's velocity at t = 0 is 5, what is the velocity of the particle at t = 3?

11.710

Limits

100

lim_h->0  (sin(x + h) - sinx)/h =

cosx

200

lim_h->0 (ln(4+h)-ln(4))/h is

1/4

300

lim_x->∞ (3x² + 5)/(6x-9x² +1) =

-1/3

400

lim_t->0 (√(1-2t) + t-1)/ t² is 

-1/2

500

lim_x->∞ (x²+2x²-6x+4)/(3x³-4x²+5x+1) is 

1/3

Random

100

Using a right Rieman sum with three subintervals and data from the table, what is the approximation of the number of liters of oil that are in the tank at time t = 15 hours?

114.9 liters

200

What is the approximation for the left Rieman sum?

39

300

If f(x)=√(x²-4) and g(x)=3x-2, then the derivative of f(g(x)) at x=3 is

7/√(5)

400

Let f(x)=(2x+1)³ and let g be the inverse function of f. Given that f(0)=1, what is the value of g'(1)?

1/6

500

What is the area of the region in the first quadrant bounded by the graph of y = e^x/2 and the line x=2?

2e-2