MTH108: Calculus II Assignment, SUSS, Singapore

Question.1

a. Evaluate ∫ 𝑒𝑒3𝑥𝑥 cos 𝑥𝑥 𝑑𝑑𝑑𝑑 .

(b) Determine the exact value of

(c) Determine the exact value of

Question.2

(a) Solve 𝑑𝑑 𝑑𝑑𝑑𝑑 ∫ sin 𝑡𝑡 𝑑𝑑𝑑𝑑sin−1 𝑥𝑥𝑒𝑒3𝑥𝑥, simplifying your answer as much as possible

b. Determine the derivative of 𝑓𝑓(𝑥𝑥) = log2021

c.Determine the derivative of 𝑔𝑔(𝑥𝑥)=

where 0 ≤ 𝑥𝑥 ≤ 𝜋𝜋 and 𝑝𝑝 > 𝑞𝑞 > 0

Question 3
(a) Interpret lim𝑛𝑛→

as a definite integral and evaluate it.

(b) Let 𝑓𝑓(𝑥𝑥) = √𝑥𝑥 − 𝑎𝑎 3 , defined over the domain 𝑎𝑎 − 8 ≤ 𝑥𝑥 ≤ 𝑎𝑎 . Determine an expression for 𝑓𝑓−1(𝑥𝑥), as well as its domain and range.

Question 4
Let 𝑅𝑅 be the region bounded by the curves 𝑦𝑦 = 𝑥𝑥2 and 𝑦𝑦 = −𝑥𝑥 + 2.
(A sketch might be helpful.)

Using the method of cylindrical shells, find the volume of the solid of revolution
obtained when 𝑅𝑅 is rotated 2𝜋𝜋 radians about the line 𝑥𝑥 = 3. Show your work clearly

Question. 5

Using the substitution 𝑥𝑥 = 2-sec 𝜃𝜃, evaluate the following integral. Show your work clearly. (In your working, you may assume that 0 ≤ 𝜃𝜃 ≤ 𝜋𝜋 2 or 𝜋𝜋 ≤ 𝜃𝜃 ≤ 3𝜋𝜋 2, which is the range of the inverse secant function.)

Question.6

Recall that a function 𝑓𝑓(𝑥𝑥) is odd if 𝑓𝑓(−𝑥𝑥) = −𝑓𝑓(𝑥𝑥) for all 𝑥𝑥 ∈ ℝ and a function 𝑔𝑔(𝑥𝑥) is even if 𝑔𝑔(−𝑥𝑥) = 𝑔𝑔(𝑥𝑥) for all 𝑥𝑥 ∈ ℝ .

(a) Suppose that 𝑓𝑓(𝑥𝑥) is an odd function. Prove that the function 𝐹𝐹(𝑥𝑥) defined by

is an even function.

(b) Use question 6(a) to show that if 𝑓𝑓(𝑥𝑥) is an odd function and 𝑐𝑐 > 0 , then

Question 7
Let 𝑓𝑓(𝑥𝑥) be a twice differentiable function on (−∞, ∞) such that 𝑓𝑓′′(𝑥𝑥) is continuous on (−∞, ∞). Further, suppose that 𝑓𝑓(0) = 𝑓𝑓(1) = 0 and that ∫ 𝑓𝑓(𝑥𝑥) 10 𝑑𝑑𝑑𝑑 = 1.
Evaluate