Name: _______________________
Date: _______________________
Class: _______________________
Chapter 5.3-7.3 Test
NO CALCULATOR
1. Find the vertical asymptote of the graph of f(x)
𝑥2 − 1
𝑓(𝑥) =
2𝑥 + 4
solution:
2. Interpret the integrand as the rate of change of a quantity and evaluate the integral using the
antiderivative of the quantity.
a.
1
∫ 𝑒 𝑥 𝑑𝑥
0
𝑒 𝑥 |10
(𝑒 1 ) − (𝑒 0 )
(𝑒) − (1) = 𝒆 − 𝟏
b.
2𝜋
∫ sin 𝑥 𝑑𝑥
𝜋
−cos 𝑥 |2𝜋
𝜋
(− cos 2𝜋) − (− cos 𝜋)
(−1) − (1) = −𝟐
3. Evaluate each integral using Part 2 Fundamental Theorem. Support your answer with NINT
if you are unsure.
1
∫ (𝑥 2 + √𝑥) 𝑑𝑥
0
𝑡
𝑡
4. Use u substitution to find the integral. ∫(1 − 𝑐𝑜𝑠 2)2 sin(2)
SOLUTION:
𝑡
𝑢 = (1 − 𝑐𝑜𝑠 )
2
1
𝑡
𝑑𝑢 = sin ( )
2
2
2 ∫ 𝑢2 𝑑𝑢
→
𝑢3
2( )
3
→
𝑡
(1 − 𝑐𝑜𝑠 (2))3
2(
)+𝐶
3
5. The function v(t) is the velocity in m/sec of a particle moving along the x-axis. Use analytic
methods to do each of the following:
a. Determine when the particle is moving to the right, to the left, and stopped
b. Find the particle’s displacement for the given time interval. If s(0) = 3, what is
the particle’s final position?
c. Find the total distance traveled by the particle
Solutions
CALCULATOR
6. What is the average value of the cosine function on the interval [1,5]?
a) -0.990
b) -0.450
c) -0.128
d) 0.412
e) 0.998
7. Evaluate each integral using Part 2 Fundamental Theorem. Support your answer with NINT
if you are unsure.
3
1
∫ (2 − ) 𝑑𝑥
𝑥
1/2
8. Use NINT to solve the problem
10
∫
0
1
𝑑𝑥
3 + 2 sin 𝑥
9. For the following problem:
a. Use Trapezoidal Rule with n=4 to approximate the value of the integral
b. Use concavity to determine if the approximation is an overestimate or an
underestimate
c. Find the integral’s exact value to check your answer
𝜋
∫ 𝑠𝑖𝑛𝑥𝑑𝑥
0
Solution
a.
b.
c.
10. find the area of the shaded region
Solution:
11. Find the volume of the solid generated by revolving the region bounded by the lines and
curves about the x-axis
a. y=x2
b. y=0
c. x=2
Solution
graph
2
𝜋 ∫ [(𝑥 2 )2 − 02 ] 𝑑𝑥
0
2
𝜋 ∫ 𝑥4
0
𝑥5 2
𝜋 ( ) |0
5
25
𝟑𝟐𝝅
𝜋 ( − 0) =
5
𝟓
12. The first three terms of an arithmetic sequence are 36, 40, 44
a.
i. write down the value of d
ii. find u8
b.
i. show that 𝑆𝑛 = 2𝑛2 + 34𝑛
ii. hence, write down the value of S14
Solution:
See Paper 2 problem 1 solution
13.
Solution:
See paper 2 problem 4 for solution