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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