AP Calculus Spring Review

Limits

100

lim_(x->oo) (x-4x^6)/(2x^5+6x^6)

-2/3

200

lim_(x->4^-)(x^2-16)/(x-4)^2

-oo

300

(dP)/dt=0.005P(2-P/1200)

The amount of plankton in a tidepool grows logistically with a growth rate function given above. What is 

lim_(t->oo)P(t)?

2400 (the A value - you have to factor out a 2 to get the correct form)

400

lim_(b->oo)int_3^b1/x^3dx

1/18

Derivatives

100

What is the instantaneous rate of change of  f(x)=(3x-1)^2 at x=2?

f'(2)=30

200

d/dt(e^(7t)tan(7t))

7e^(7t)(tan(7t)+sec^2(7t))

300

Write the equation of the line tangent to the curve below at x=4

y=sqrtx/(x-2)

y-1=-3/8(x-4)

400

f(2)=4, f'(4)=3, g(2)=4, g'(2)=-1, g'(4)=2

Find the derivative of f(g(2x)) at x=1

-6

Applications

100

What is the formula for the average value of a function f(x) from x=a to x=b?

1/(b-a)int_a^bf(x)dx

200

If  f'(x)=x and 

f(2)=1, then f(4)=

7 (SPAM)

300

The Mean Value theorem promises that if a function is differentiable and continuous on (a,b) then there is a point c where f'(c)=

(f(b)-f(a))/(b-a)

400

Sand falls into a conical pile in such a way that the height of the pile increases at 3in/min. The radius of the cone is always half the height of the cone. If the volume of a cone is given by  V=1/3pir^2h , what is 

 (dV)/dt when the height is 40in?

Include units.

1200pi "in"^3/min

Integration

100

int(sin(2x)dx

-1/2cos(2x)+c

200

int(lnx/x)dx

(lnx)^2/2+c

300

inte^cscx(cscxcotx)dx

-e^cscx+c

400

int_0^6 2/(9x^2-1)dx

1/3ln(17/19)

BC only

100

Use Euler's Method with a step size of 0.5, starting at (1,2) to approximate f(2), for the function f whose derivative is given by 

dy/dx=x-2y

1/4

200

The equation below will be a limacon with a loop if what condition is met? (Assume k and p are positive numbers)

 r=k+p sin(theta)

p>k

300

What is the radius of convergence for 

sum_(n=1)^(oo) (x-2)^n/(5^nsqrtn)

x in [-3, 7)

400

What is the coefficient of the 8th degree term in the Taylor series for 

x^2cos(x^3)

-1/2