2 hours 30 minutes90 marks

Level 1 - Calculus Paper 1

A full Level 1 - Calculus exam covering limits, differentiation, applications of derivatives, integral methods, series, multivariable calculus, and differential equations.

Instructions

Attempt all questions.
All main questions carry 15 marks.
Show sufficient working to justify each answer.
Exact answers are preferred unless a decimal approximation is requested.

Section A: Limits and Continuity

1. Limits, continuity, and graph behaviour

15 marks

This question concerns limiting behaviour, continuity, and asymptotes of single-variable functions.

(a)

4 marks

Evaluate \(\displaystyle \lim_{x\to4}\frac{\sqrt{x+5}-3}{x-4}\).

(b)

4 marks

Let \(f(x)=ax+2\) for \(x<1\), and \(f(x)=x^2+b\) for \(x\ge1\). Find the relationship between the constants \(a\) and \(b\) required for \(f\) to be continuous at \(x=1\).

(c)

3 marks

Find the vertical and horizontal asymptotes of \(g(x)=\frac{2x+1}{x-2}\).

(d)

4 marks

Using first principles, find the derivative of \(f(x)=x^2-3x\).

Section B: Differential Calculus

2. Derivative methods

15 marks

This question uses differentiation rules, implicit differentiation, and local approximation.

(a)

4 marks

Differentiate \(f(x)=(x^2+1)e^{-x}\).

(b)

4 marks

For the implicit curve \(xy^2+x^2=5\), find \(\frac{dy}{dx}\).

(c)

3 marks

For \(f(x)=x^4-6x^2+3\), find \(f''(2)\).

(d)

4 marks

Use linear approximation about \(x=9\) to estimate \(\sqrt{9.3}\).

Section C: Applications of Derivatives

3. Using derivatives to answer applied questions

15 marks

This question concerns optimisation, related rates, curve information, and indeterminate limits.

(a)

4 marks

A rectangle has side lengths \(x\) and \(y\), with \(x+y=12\). Find the side lengths that maximise its area.

(b)

4 marks

A sphere has radius \(r\) increasing at \(0.5\) units per second. How fast is the volume increasing when \(r=2\)?

(c)

4 marks

For \(f(x)=x^3-6x^2+1\), find and classify the local stationary points.

(d)

3 marks

Use L'Hopital's rule to evaluate \(\displaystyle \lim_{x\to0}\frac{e^{2x}-1}{x}\).

Section D: Integral Calculus

4. Integral methods

15 marks

This question uses antiderivatives, substitution, integration by parts, partial fractions, and the fundamental theorem of calculus.

(a)

3 marks

Evaluate \(\displaystyle \int 2x(x^2+4)^5\,dx\).

(b)

4 marks

Evaluate \(\displaystyle \int x\cos x\,dx\).

(c)

4 marks

Evaluate \(\displaystyle \int\frac{5x+6}{x^2-4}\,dx\), for \(x\ne\pm2\).

(d)

4 marks

Differentiate \(\displaystyle A(x)=\int_1^{x^2}\ln(1+t^2)\,dt\).

Section E: Applications of Integrals and Series

5. Integral applications and infinite processes

15 marks

This question combines accumulated area, volumes, improper integrals, and series convergence.

(a)

4 marks

Find the area enclosed by \(y=x\) and \(y=x^2\).

(b)

4 marks

Find the volume formed by rotating the region under \(y=\sqrt{x}\) from \(x=0\) to \(x=4\) about the \(x\)-axis.

(c)

4 marks

Determine whether \(\displaystyle \int_2^\infty \frac{1}{(x-1)^2}\,dx\) converges, and evaluate it if it does.

(d)

3 marks

Use the ratio test to determine whether \(\displaystyle \sum_{n=1}^{\infty}\frac{3^n}{n^2}\) converges.

Section F: Multivariable Calculus and Differential Equations

6. Several variables and differential equations

15 marks

This question concerns functions of several variables and elementary differential equations.

(a)

4 marks

For \(f(x,y)=x^2y+3xy^2\), find \(\nabla f(1,2)\).

(b)

3 marks

Using the function from part (a), find the directional derivative at \((1,2)\) in the unit direction \(\mathbf u=(3/5,4/5)\).

(c)

4 marks

Let \(z=f(x,y)=x^2y+3xy^2\), where \(x=t\) and \(y=t^2\). Find \(\frac{dz}{dt}\) at \(t=1\).

(d)

4 marks

Solve \(\frac{dy}{dx}=2xy\) with initial condition \(y(0)=3\).