Methods2024 NHT Exam 2 Section B · Q1CAS active10 marks

Cubic with a parameter: intercepts, turning points, tangent and area

Consider the function $f: \R \to \R, f(x) = x^3 - px$ , where .

Find the values of the $x$ -intercepts of $f$ , when $p = 3$ .

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

Use the derivative $f'$ to find the coordinates of the turning points of $f$ , when $p = 3$ .

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

c.i.

Find the value of $p$ for which $f$ would have exactly one stationary point.

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

c.ii.

Find the values of $p$ for which $f$ would not have any stationary points.

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

d.i.

The graph of $f$ passes through the origin for all values of $p$ .

Use calculus to show that the tangent line to $f$ at the origin has the equation $y = -px$ .

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

d.ii.

Find, in terms of $p$ , the area of the region bounded by the function $f$ , the line $y = -px$ and the line $x = p$ , where $p > 0$ .

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

Methods2024 NHT Exam 2 Section B · Q1CAS active10 marks

Cubic with a parameter: intercepts, turning points, tangent and area

Consider the function $f: \R \to \R, f(x) = x^3 - px$ , where .

Find the values of the $x$ -intercepts of $f$ , when $p = 3$ .

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

Use the derivative $f'$ to find the coordinates of the turning points of $f$ , when $p = 3$ .

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

c.i.

Find the value of $p$ for which $f$ would have exactly one stationary point.

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

c.ii.

Find the values of $p$ for which $f$ would not have any stationary points.

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

d.i.

The graph of $f$ passes through the origin for all values of $p$ .

Use calculus to show that the tangent line to $f$ at the origin has the equation $y = -px$ .

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

d.ii.

Find, in terms of $p$ , the area of the region bounded by the function $f$ , the line $y = -px$ and the line $x = p$ , where $p > 0$ .

2 marks

Hints, answer checking and worked solutions come with a Season Pass.