AP Calculus Project

Limits

100

The asymptote of lim _{x -> -∞} e^x - 5

What is -5

200

   The asymptote of
             5x^3 +3x + 7
lim _{x -> ∞} ----------------
             2x^2 + 5x - 2

What is ∞

300

Suppose lim _x->2 f(x) = 4, lim _x->2 g(x) = 5, lim _x->2 h(x) = 8. The asymptote of lim _x->2 [6f(x)g(x) + h(x)]

What is 128

400

Using conjugates, the approaching value at lim _{x-> 1} (√x -1)/(x-1)

What is 1/2

500

          5(x²-4)
f(x) = ----------
         x²+2x-8

lim _{x -> ∞} f(x) and all the horizontal asymptotes of f

what is lim _{x -> ∞} f(x) = 5 and y = 5

Derivatives

100

The names of the derivatives of position in order, after acceleration

What is jerk, snap, crackle, pop

200

The formula for d/dx (csc^-1(u))

            -u'
What is --------------
           |u|√(u²-1)

300

                                x³ - 3x² +4
The value as lim _{x -> 2} ----------------------------
                                x⁴ - 4x³ + 7x² - 12x + 12

             6    3
What is ---  ---
            14,  7

400

The X value for the absolute minimum of the function f(x) = x²ln(x) on the domain (0, ∞)

What is x = e^-1/2

500

Two small planes approach an airport, one flying due west at 120 mi/hr and the other flying due north at 150 mi/hr. Assuming flying at constant elevation, how fast is the distance between the planes changing when the westbound plane is 180 miles from the airport and the northbound plane is 225 miles from the airport?

What is dc/dt = -192/hr

Integrals

100

The antiderivative of

          1
f(x) = ------------
          1 + x²

What is tan^-1(x) + C

200

The antiderivative of ∫(2x+3)³ dx 

What is (1/8)(2x+3)⁴ + C

300

The area between the curves f(x) = 5 - x² and g(x) = x² - 3

What is 64/3

400

The volume of a solid bounded by y = x² and y = 2 - x² using semi-circles perpendicular to the x-axis

What is 8π/15 ≈ 1.6755

500

Let R be the region bounded by the curve f(x) = (x+1)², the x-axis, the lines x = 0 and x= 2. The volume of the disk created by revolving R around the x-axis

What is 242π/5 ≈ 152.0530

Random

100

Does the IVT apply to the function f(x) = x² - x + 1 on [-1, 5] where f(c) = 8

Yes as f(c) falls between f(-1) and f(5)

200

The x-value where the instantaneous rate of change equals the average rate of change on the interval [2, 6] for the function f(x) = -x² + 3x + 10

What is x = 4

300

The general solution for the differential equation
dy     x²
--- = ---
dx     y

y = ∓√((2/3)x³ + c)

400

The total distance traveled by a jogger on the interval [0, 3] whose velocity is given by v(t) = 2t² - 8t + 6 miles/hour

What is 16/3 ≈ 5.333

500

The function that gives the total amount of cells in a culture in a lab when the growth rate is modeled by N'(t) = 90e^-0.1t cells/hr and and N(0) = 100

What is N(t) = N(0) + ∫90e^-0.1x dx [0, t]