91578 – Worked Solutions - 2025

QUESTION THREE

3.6 (AS 91578) – 2025 Answers

Quite often, there are different ways to solve a Mathematics problem.

These solutions/strategies show one possible way to solve them.

​

Question 3(a)

​

Question 3(b)

​

Question 3(c)

​

Question 3(d)

​

Question 3(e)

Question 3(a)

Question 3(a)

Question 3(a)

A function is NOT differentiable when

• Discontinuous (breaks in the function)
• A sharp change in gradient, without going through a stationary point
• Vertical line

Question 3(a)

There are three ‘breaks’ in this function.

An open circle indicates the value at that point does not exist.

A function is NOT differentiable when

• Discontinuous (breaks in the function)
• A sharp change in gradient, without going through a stationary point
• Vertical line

Question 3(a)

There are three ‘breaks’ in this function.

An open circle indicates the value at that point does not exist.

Question 3(a)

Question 3(a)

Question 3(a)

Stationary Points

Question 3(a)

These are NOT Stationary Points

Question 3(a)

Stationary Points

Question 3(a)

Stationary Points

Zero Gradient

Question 3(a)

Stationary Points

Zero Gradient

Question 3(a)

Question 3(a)

Question 3(a)

Question 3(a)

Question 3(a)

Question 3(b)

Question 3(b)

Question 3(b)

Question 3(b)

We can use the quotient rule to do this.

Question 3(b)

Question 3(b)

Question 3(b)

Question 3(b)

Question 3(b)

Question 3(b)

Question 3(b)

Question 3(b)

Question 3(b)

Question 3(b)

Simplify

Question 3(b)

Question 3(b)

Question 3(b)

Question 3(b)

Divide both sides by 2

Question 3(b)

Question 3(b)

Question 3(c)

Question 3(c)

Rate simply means gradient of the volume with respect to time

Question 3(c)

Volume of a sphere

Question 3(c)

Differentiate the volume with respect to the radius, r and simplify.

Question 3(c)

Question 3(c)

Question 3(c)

Question 3(c)

Question 3(c)

Question 3(c)

Question 3(c)

Question 3(c)

Question 3(c)

Question 3(c)

Question 3(c)

Question 3(d)

Question 3(d)

Question 3(d)

This is a parametric function.

​

We need to differentiate this to get a gradient function.

Question 3(d)

Question 3(d)

Question 3(d)

Question 3(d)

Question 3(d)

Question 3(d)

Question 3(d)

Question 3(d)

Question 3(d)

Question 3(d)

Question 3(d)

Question 3(d)

Question 3(d)

Question 3(d)

Question 3(d)

Gradient Function

Question 3(d)

Gradient Function

Question 3(d)

Question 3(d)

Question 3(d)

Question 3(d)

Divide both sides by 4

Question 3(d)

Question 3(d)

Question 3(d)

Question 3(d)

Question 3(d)

Question 3(d)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Rearrange to get equal to zero

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

This is the equation of a circle, with a shift of 8 horizontal squares to the right and no vertical shift, and a radius of 8.

Question 3(e)

Diameter of the circle = 16

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

To maximise a function (eg find the maximum area), we need to differentiate the area function and put this gradient function equal to zero to find a maximum turning point.

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Expand brackets and get quadratic equal to zero

Question 3(e)

Solve quadratic on calculator or by quadratic formula

Question 3(e)

Solve quadratic on calculator or by quadratic formula

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)

Question 3(e)