Assignment 10

AMERICAN UNIVERSITY OF THE MIDDLE EAST

MA166 – Analytic Geometry and Calculus II, Fall 2020, Section: O2

Assignment 1, Due: Nov 17th

Instructions: There are 4 questions. Please answer clearly, show all your work step by step.

Student Name/ID: ___________________________________________________________ Student Name/ID: ___________________________________________________________ Q1 [25 pts]: Justify your answers clearly.

a) [10 pts] Find the derivative of the function 𝑔(𝑥) = ∫ 1

𝑡 𝑑𝑡

𝑥

1 , 𝑥 > 0

b) [8 pts] Evaluate the definite integral ∫ (1 + cos𝑥)𝑑𝑥 𝜋

0

c) [7 pts] Let 𝑔(𝑥) = ∫ 𝑓(𝑡)𝑑𝑡 𝑥

0 , 0 ≤ 𝑥 ≤ 5, where 𝑓 is the function whose graph is shown

below.

Specify the point on the graph where the value of 𝑔(𝑥) is minimum. Explain your answer.

Q2 [25 pts]: Justify your answers clearly.

Suppose that ∫ 𝑓(𝑥)𝑑𝑥 2

−2 = 4, ∫ 𝑓(𝑥)𝑑𝑥

5

2 = 3, ∫ 𝑔(𝑥)𝑑𝑥

5

−2 = 2

a) [10 pts] Find ∫ 𝑓(𝑥)𝑑𝑥 5

−2

b) [8 pts] Find ∫ (𝑓(𝑥) + 𝑔(𝑥))𝑑𝑥 5

−2

c) [7 pts] Is it true that if 𝑓(𝑥) is a continuous function on the interval [𝑎,𝑏] such that

∫ 𝑓(𝑥)𝑑𝑥 𝑏

𝑎 ≥ 0, then 𝑓(𝑥) ≥ 0 on [𝑎,𝑏]? Justify your answer.

Q3 [25 pts]: Justify your answers clearly.

a) [10 pts] Evaluate the definite integral ∫ 3

(𝑥+3)4 2

−2 𝑑𝑥

b) [8 pts] Evaluate the indefinite integral ∫2(2𝑥 + 4)5𝑑𝑥

c) [7 pts] Explain, without calculating the areas, why the area under the graph of the curve 𝑦 =

2(2𝑥 + 4)5 from 𝑥 = 0 to 𝑥 = 3 is equal to the area under the graph of the curve 𝑦 = 𝑥5

from 𝑥 = 4 to 𝑥 = 10.

Q4 [25 pts]: Justify your answers clearly.

a) [10 pts] Evaluate the indefinite integral ∫ 𝑥

1+𝑥 𝑑𝑥

b) [8 pts] Evaluate the definite integral ∫ (𝑥 + 1)4𝑑𝑥 2

−1

c) [7 pts] What is the meaning of the integral in Part (b) in terms of areas?