Calculus Answers

Differentiate with respect to 𝑥

a. (7𝑥 – 4 )³

b. √(6𝑥+4)

The curve 𝑦=−𝑥³+3𝑥²+6𝑥−8 cuts the 𝑥-axis at 𝑥=−2,𝑥=1 and 𝑥=4.

a. Sketch the curve, showing clearly the intersection with the coordinate axes.

b. Differentiate 𝑦=−𝑥³+3𝑥²+6𝑥−8

c. Show that the tangents to the curve at 𝑥=−2 and 𝑥=4 are parallel.

A curve is described by 𝑦=𝑝𝑥²+𝑞𝑥, where 𝑝 and 𝑞 are constants.

a. Find an expression for the gradient of this curve at any point.

b. Given that at the point (1,−2) the gradient is 6, calculate the values of 𝑝 and 𝑞.

c. Show that the equation of the normal to the curve at the point (1,−2) can be written as 𝑥+6𝑦+11=0.

Find (x̅, y̅): R = {(x, y): 0 ≤ y ≤ √x^2 + 1 , 0 ≤ x ≤ 1} about the x-axis.

Find the center of mass of the solid generated by the area bounded by x = 1, x = 3, y = 0

and y = x^2 by revolving about the x-axis.

Using shell method, find the volume of R, when it is bounded by √x + √y = √a , x =

0 , y = 0 about the line x = a.

Differentiate:

f(x)= (2x⁴+5x+2)

g(x)= 3x²√6x³+5x²+1

h(x)= 4x²/√x+7

find the surface area of the portion of the curve x^2+y^2=4 from x=0 to x=2 when it is revolved about the y-axis

Topic: Implicit Differentiation

1. Find y’ in 𝑥3 + 2𝑦3 = 3𝑥2𝑦.

2. Find the derivative of 𝑦 = √𝑠𝑖𝑛𝑥𝑦

Topic: Optimization

1. A close right circular cylinder is to be constructed to hold a 1 liter oil can shape.

What dimensions will minimize the amount of material, assuming that the

thickness of the material is uniform?

2. Find two positive numbers whose sum is 9 and whose product is a maximum.