AP Calculus
Third Quarter Exam Review
Unit 5B: Definite Integrals (Sections 5.6 – 5.9)
2
 Find definite integrals by hand using the Fundamental Theorem of Calculus: ∫1 √4𝑥 − 2 𝑑𝑥
 Find definite integrals using a graphing calculator
 Use definite integrals to calculate areas of regions bounded by functions.
1.) Calculate the area between x = 0 and x = 𝜋,
bounded by y = sin x and the x-axis.
2.) Determine the area of the region bounded by
𝑦 = 𝑥 2 − 2𝑥 − 2 and y = -x + 4.
3.) Calculate the area of the region bounded by
𝑥 = 𝑦 2 + 2𝑦 − 8 and the y-axis.

Use definite integrals to calculate displacement given a velocity function.
4.) A radio-controlled car’s velocity in m/s is defined as a function of time after t seconds by the function 𝑣(𝑡) = 2√𝑡 𝑑𝑡.
To the nearest tenth of a meter, how far did it travel from t = 2 to t = 8 seconds?

Use definite integrals to calculate volumes of solids of rotation (disc method and washer method) around the x
and y axes. (Remember that the area of one cylindrical disc is 𝑑𝑉 = 𝜋 𝑟 2 ℎ )
5.) Region bounded by:
y = 𝑒 𝑥 , x-axis, x = 1, x = 5
6.) Region bounded by:
𝑦 = 4√𝑥, 𝑦 = 2𝑥
7.) Region bounded by:
𝑦 = 4 − 𝑥 2 , the x-axis
Rotated around x-axis.
Rotated around x-axis
Rotated around the y-axis.
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Unit 8: Maximum/Minimum Values of Functions (Sections 8.2 – 8.3)
 Find critical values of function (where derivative = 0 or is undefined)
 Draw first and second derivative number lines (showing +/- slope and up/down concavity)
 Create a graph of a function given first/second derivative number lines
 Identify relative (local) max (derivative changes from + to -) and minimums (der. changes from - to +)
 Find absolute (global) max/min values (check endpoints of interval in addition to relative max/min)
 Identify plateau points (critical value, but no change in direction)
 Identify points of inflection (second derivative changes sign… concavity change)
 Determine rel max/min of function using the Second Derivative Test
 Solve max/min word problems
12.) Use the first and second derivative number lines to sketch function f.
13.) Given the function 𝑦 = 3𝑥 4 − 16𝑥 3 + 18𝑥 2 :
A) Find the first derivative and calculate the critical values.
B) Use your answers from part A to create a first derivative number line.
C) Based on your number line, at what values of x does a local max occur? A local min?
D) What is the absolute maximum of the function on the interval [-1, 4]?
E) Find the second derivative of the function and create a second derivative number line.
F) For what intervals of x is the original function concave up? Concave down?
G) For what values of x are there points of inflection?
H) Use your number lines to sketch the original function.
14.)
15.) Function f has a critical point where x = 1. If f”(1) = -8, does the function have a relative max or min at x = 1?
Unit 6: Calculus of Exponential and Logarithmic Functions (sections 6.2 – 6.6)
 Take derivatives of exponential and logarithmic functions.
16.) Determine each derivative:
𝑑
A) 𝑑𝑥 52𝑥