Calculus Answers

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.

explain clearly.

∫(7𝑐𝑠𝑐2𝑥 + 2 sec 𝑥 tan 𝑥)𝑑𝑥

1. Apply your mathematical models to your allocated car. Use the given data for the 0 – 28 m/s and 400m times to calculate the:

1. value of the coefficient A

2. maximum velocity

3. maximum acceleration.

Given: t (0-28 m/s) is 1.9s, t (400m) is 10.50s & tmaxspeed is 7.1s.

Given velocity equation :V (t) = A (1 - e^((-t)/(t maxspeed)))

i need step by step solution to understand