AP Calculus AB Semester 2

Analytical Applications of Derivatives

100

Let f be the function f(x) = x²-6x-7. Find all local extrema in the interval [0,10].

x = 3 ONLY

200

Let f be the function defined by f(x)=x³−12x²+21x+4 for 0<x<5. On what interval between (0,5) is f increasing? Give a reason why?

f is increasing on (0,1) since f’(x)>0 on that interval

300

CALCULATOR ALLOWED

Let f be the function 3x³+7x²-9x-9, what is the absolute maximum value of this function on the interval [-5,5].

The absolute maximum value is 356 at x=5.

400

Let f be the function x³-4x²-3x-7, on what interval is the graph of f concave up and increasing on the interval [-5, 5].

(3,5), since f’(x)>0 on (3,5), and f”(x)>0 on (4/3,5).

500

Find two numbers whose sum is 324 and whose product is as large as possible.

162 & 162

Integrals

100

∫sin x dx =

-cos x + C

200

CALC ALLOWED

∫₃¹⁴(x²-13x+42) dx =

152.166/152.167

300

∫4x³-21x²+42x+3 dx =

x⁴-7x³+21x²+3x+C

400

Approximate the area under the graph of x³ using a Right Riemann sum approximation from x=0 to x=4 with 4 equal subintervals?

100

500

CALC ALLOWED

If function h(x) = the derivative of the integral of 3x²-12x+30 bounded by the lower limit 0 and the upper limit x, what is the value of h(7)?

93

Differentials

100

Solve ∫cos (6x-5) dx

1/6[sin(6x-5)]+C

200

Solve ∫16/(x²-14x+50) using completing the square

16tan^-1(x-7)+C

300

Find the particular solution that passes through the given point (2,0). dy/dx = e^x-24x²

e^x-8x³+64-e²

400

Of the following, which are solutions to the differential equation 4y+y′′=0 ?
1. y=3sin(2x)
2. y=5cos(2x)
3. y=e^2x

1 and 2

500

dA/dt=1/4(160-A)

Use separation of variables to find y = A(t), the particular solution to the differential equation with initial condition A(0) = 40.

A(t) = 160 - 120e^-1/4t

Applications of Integrals

100

If a particle moves along the x axis, and its position is given by x(t), and it’s velocity given by v(t). How would I find the TOTAL distance traveled over a given interval between t=3 and t=9.

∫|v(t)| bounded by the upper (9) and lower limits (3) of the interval.

200

CALC ALLOWED

Let f be the function given by f(x) = 6sin(x)cos(2x). What is the average value of f on the closed interval [5,8]?

-0.823/-0.824

300

CALC ALLOWED

The rate at which people enter a grocery store is E(t) = 10050/(t²-34t+160) during the period t=0 to t=12, with t representing time after 10AM.

How many people have entered the grocery store by 3PM (t=5)? Round you answer to nearest whole number.

876 people

400

CALC ALLOWED
Find the area of the region enclosed by y = x²+2x and y=x, and the x-axis, from x=0 to x=6.

90

500

CALC ALLOWED
Determine the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis: y=x, y=0, x=3, x=6.

197.920