Calculus Answers

Let (fn) be defined by ∀n ∈ N

f : [0,π/2] → R

x → n(cos(x)^n)(sin(x))

1. Show that f has at most one maximum value at a point xn. (xn is a maximum point of fn(x))

2. Show that xn ∼√n

A ladder, inclined at 60 ° with the horizontal is leaning against a vertical wall. The foot of the ladder is 3 meters away from the foot of the wall. A boy climbs the ladder such that his distance z meters with respect to the foot of the ladder is given by z = 6t, where tis the time in seconds. Find the rate at which his vertical distance from the ground changes with respect to ¢. Find the rate at which his distance from the foot of the wall is changing with respect to t when he is 3 m away from the foot of the ladder.

Provide all necessary steps and evaluate the following integrals:

(a) ∫ (𝑥^2√(2 + 𝑥)) 𝑑𝑥

(b)∫ (2^𝑡/(2^𝑡 + 3)) 𝑑𝑡

(c) ∫ 𝑑𝑡/(cos^2(𝑡 √1 + tan 𝑡))

Show that the minimum and maximum points of every curve in the family of polynomials

𝑓(𝑥) = 2𝑥^3 + 𝑐𝑥^2 + 2𝑥 lie on the curve 𝑦 = 𝑥 − 𝑥^3

.

A trough whose cross-section is an equilateral triangle which is 6 m long and 2 m wide across the top. If water is entering the trough at 15 m³/min, at what rate is the water level rising in the trough when it is three- fourths full?