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AP Calculus Chapter 5 Test—Alternate
Name: Date: Period:
Part I. Multiple-Choice Questions
- Consider the region R below bounded by the curves x = 5 − y^2 , x = 1, and y = 1.
Place graph1alt here
Now consider the statements
I. Area of R =
1
5 − x − 1) dx
II. Area of R =
1
(4 − y 2 ) dy
III. Area of R =
1
(x 2 − 4) dx
(A) Only I is true.
(B) Only II is true.
(C) Only III is true.
(D) I and II are true.
(E) I and III are true.
d
dx
∫ (^) x 2
−x^2
t
4 − t^2 dt =
(A) 2 t^3
4 − t^4
(B) 2 x 3
4 − x^4
(C) 2 x^2
4 − x^4
(D) 0
(E) − 2 x 3
4 − x^4
∫ π 4
0
sin x dx +
− π 4
cos x dx =
(A) −
2 (B) − 1 (C) 0 (D) 1 (E)
- If
30
f (x) dx = 10 and
50
f (x) dx = A, then
30
f (x) dx =
(A) A + 10 (B) A − 10 (C) 0 (D) 10 − A (E) 4 A
- The average value of the function f (x) = 1 + ln 2 x on the interval [2, 4] is
(A) − 0. 204 (B) 2. 204 (C) 3.159 (D) 3. 408 (E) 6. 636
2 x ln 2x dx =
(A)
ln 2x
x
+ C
(B)
ln 2x
2 x
+ C
(C) x ln 2x − x + C
(D) x 2 ln 2x − 1 2 x
2
(E) 2 x ln 2x − 2 x + C
- Approximate
0
sin 2 x dx using the Trapezoid Rule with n = 4 to three deci-
mal places.
(A) 0.277 (B) 0.131 (C) 0.124 (D) 1.109 (E) 2.
Part II. Free-Response Questions
- Consider the ellipse whose equation is given by
(x − 2)^2
4
(a) (5 points) Set up a definite integral which will compute the area inside this ellipse.
(b) (5 points) Use your calculator to compute this area. (You may sketch any pictures as needed on the axes below.)
-
6
?
x
y
- (5 points) Compute the positive value of a in the equation below below (where
a > 0 ): ∫ (^2) a
a
(x + 2) dx = 10
- Compute (5 points)
x + 1 √ x
dx
Place pwlinear
- Let f be the function defined on the closed interval [0, 7]. The graph of f ,
consisting of four line segments, is shown above. Let g be the function given
by g(x) =
∫ (^) x
2
f (t) dt.
(a) (3 points) Find g(3), g ′ (3), and g ′′ (3).
(b) (3 points) Find the average rate of change of g on the interval 0 ≤ x ≤ 3.
(c) (4 points) For how many values c, where 0 < c < 3 , is g ′ (c) equal to the average rate found in part (b)? Explain your reasoning.
Place graph
- Let f be the function given by f (x) = 4x^2 −x^3 , and let L be the line y = 18− 3 x,
where L is tangent to the graph of f. Let R be the region bounded by the graph of f and the x-axis, and let S be the region bounded by the graph of f , the line l, and the x-axis, as shown above.
(a) (5 points) Show that L is tangent to the graph of y = f (x) at the point x = 3.
(b) (5 points) Find the area of S.
