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Calculus II Test #1 March 4, 2005
Name____________________ R. Hammack Score ______
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(1) Find the following antiderivatives.
(a) ∫( 3+
+ln(3) )dx = 3
+ln|x|+x ln(3)+C
(b) ∫ x dx = ∫ x
dx =∫
dx =
+C =
+C
(c) ∫ dx =5
(x)+C
(d) ∫( +sin(π x) + sec(x)tan(x) )dx = -
-
cos(π x) + sec(x)+C
(e) ∫sin(3x) dx =
-∫
(-1)sin(3x)3 dx =-
∫
du =-
+C=-
+C
u = cos(3x) dx
du = -sin(3x)3 dx
(f) ∫xdx =
∫(u+3)du =∫(
+3
)du =
+3
+C=
+2
+C=
+2
+C
u = x - 3
du = dx
(2) Find the following definite integrals. Simplify your answer as much as possible.
(a) (x) dx =tan(π/3)-tan(π/4)=
-1
(b) 3
dx =3
-3
=6-3=3
(c) (
+x-1)dx =
=
+
-1=-
(d) cos (
) 2x dx =
cos (u) du =sin (π/2)-sin (0)=1
u =
du = 2x dx
(e) dx =
2x dx =
du =
=
(-
+
)=-
u = +2
du = 2x dx
(f) dx =
du =
=ln|2e|- ln|e|=ln(2)+ln(e)-ln(e)=ln(2)
u = x +e
du = dx
(3) Consider the limit of Riemann sums
(
+1)Δ
, where a=1 and b=2.
Find the value of this limit by writing it as a definite integral and evaluating.
(
+1)dx =
=(
+2)-(
+1)=
+2-
-1=
(4) Suppose a particle moving on the number line has a velocity of v(t)=t-1 units per second at time t.
(a) Find the object's displacement between times t = 0 and t = 4.(t-1)dt =
=
-4=4 units
(b) Find the total distance the object has traveled between times t = 0 and t = 4.|t-1|dt =
|t-1|dt +
|t-1|dt =
(1-t)dt +
(t-1)dt=
+
=5units
(5) This question concerns the function F(x)=
dt.
(a) F '(x)=
(b) Find the interval(s) on which F(x) increases.
F '(x)= =
=-
From this you can see the derivative is positive on the interval [4,5], so that is where F increases.
(6) Find the area of the region contained between the x-axis and the graph of y=1-.
y=1-=(1-x)(1+x), so the x-intercepts are 1 and -1.
(7) Suppose
f(x) ={
|
2 if x <0 |
2-x if x≥0 |
Integral equals area of a 3 by 2 rectangle plus area of triangle of height and base 2.
Thus its value is (3)(2)+1/2(2)(2) = 8
(b) Find a value of x, different from -3, for which f(x)dx = 0
Notice that from the graph, if x = 6, the net signed area is 0. Hence the answer is x = 6.