# Exam: [Course code] [Exam name] with Answer Key

> Exam paper template with a separate answer key: candidate instructions,
> multiple-choice, short-answer and extended problem sections, a point
> allocation table, worked solutions written in KaTeX math, and marking
> guidance. The sample questions use introductory calculus and algebra so
> the format is concrete; replace them with your own. Based on common
> assessment practice.

<!--
Guidance: How to use this template
- Build the point allocation table (section 2) first, from the learning
  outcomes, so the exam samples the course fairly.
- Write the answer key at the same time as the questions. Writing the
  solution often reveals an ambiguous or too-long question.
- Keep the answer key in a separate file or remove it before printing the
  student version. Export each version separately.
- Have a colleague sit the exam with the key hidden; their time should be
  well under the time allowed (a common rule is about half).
-->

| Field | Value |
|---|---|
| Course | [MATH 110 Calculus I] |
| Exam | [Midterm exam 1] |
| Date / time | YYYY-MM-DD, [09:00-10:30] |
| Duration | [90 minutes] |
| Total points | [50] |
| Permitted materials | [Non-programmable calculator; one A4 formula sheet] |
| Examiner | [@name] |
| Moderator | [@name] |
| Version | [A] |

## 1. Instructions to candidates

<!-- Guidance: Clear, short instructions printed on the cover page. State what to do when a question seems unclear. -->

1. Write your name and student ID on every page.
2. Answer **all** questions in Sections A, B and C.
3. In Section A, circle one answer per question. No working is required.
4. In Sections B and C, show all working. Correct answers without working may
   receive limited credit.
5. Give exact answers unless a question asks for a decimal approximation.
6. If you believe a question contains an error, state your assumption and
   answer the question as you interpret it.
7. Points for each question are shown in brackets.

| Name | Student ID | Seat |
|---|---|---|
| [ ] | [ ] | [ ] |

## 2. Point allocation

<!-- Guidance: Show how points map to sections and learning outcomes. Check that section totals sum to the exam total. -->

| Section | Questions | Points each | Section total | Learning outcomes |
|---|---|---|---|---|
| A. Multiple choice | 1-5 | 2 | 10 | LO1, LO2 |
| B. Short answer | 6-8 | 4 | 12 | LO2, LO3 |
| C. Problems | 9-10 | 14 | 28 | LO2, LO3, LO4 |
| **Total** | **10** | | **50** | |

Check: 10 + 12 + 28 = 50.

## 3. Section A: Multiple choice (10 points)

<!-- Guidance: One correct answer and three plausible distractors based on common errors. Avoid "all of the above" and negatively worded stems. -->

**Q1.** [2] What is $\dfrac{d}{dx}\left(x^3\right)$?

(a) $x^2$ &nbsp; (b) $3x^2$ &nbsp; (c) $3x^3$ &nbsp; (d) $\dfrac{x^4}{4}$

**Q2.** [2] Evaluate $\displaystyle\int_0^3 2x \, dx$.

(a) 3 &nbsp; (b) 6 &nbsp; (c) 9 &nbsp; (d) 18

**Q3.** [2] What is $\displaystyle\lim_{x \to 0} \frac{\sin x}{x}$?

(a) 0 &nbsp; (b) 1 &nbsp; (c) $\infty$ &nbsp; (d) The limit does not exist

**Q4.** [2] What is $\dfrac{d}{dx}\left(e^{2x}\right)$?

(a) $e^{2x}$ &nbsp; (b) $2e^{x}$ &nbsp; (c) $2e^{2x}$ &nbsp; (d) $2x\,e^{2x-1}$

**Q5.** [2] What is the slope of the tangent to $y = x^2$ at $x = 3$?

(a) 3 &nbsp; (b) 6 &nbsp; (c) 9 &nbsp; (d) 2

## 4. Section B: Short answer (12 points)

<!-- Guidance: Questions answerable in a few lines. State exactly what is required: a value, a statement, a classification. -->

**Q6.** [4] Use the product rule to differentiate $f(x) = x^2 \sin x$.

**Q7.** [4] Find the critical points of $f(x) = x^3 - 3x$ and classify each as
a local maximum or local minimum.

**Q8.** [4] Solve $2x^2 - 8x + 6 = 0$.

## 5. Section C: Problems (28 points)

<!-- Guidance: Multi-part problems that assess reasoning. Order parts from routine to demanding, and make later parts answerable even if an earlier part is wrong (give intermediate values where needed). -->

**Q9.** [14] Let $f(x) = \left(3x^2 + 1\right)^4$.

(a) [6] Find $f'(x)$.

(b) [3] Evaluate $f'(1)$.

(c) [5] Find the equation of the tangent line to $y = f(x)$ at $x = 0$.

**Q10.** [14]

(a) [8] Find the area of the region enclosed by $y = x^2$ and $y = 2x$.

(b) [6] Solve the system of equations
$$
\begin{cases} 2x + y = 7 \\ x - y = 2 \end{cases}
$$

---

## 6. Answer key

<!-- Guidance: Remove this section from the student version. Give the final answer and a full worked solution for every question, with the marks attached to each step. -->

### 6.1 Section A

| Question | Answer | Common wrong answer and the error it reveals |
|---|---|---|
| Q1 | (b) $3x^2$ | (d): integrated instead of differentiated |
| Q2 | (c) 9 | (d): forgot to divide by 2 when integrating |
| Q3 | (b) 1 | (d): treated $0/0$ as undefined without taking the limit |
| Q4 | (c) $2e^{2x}$ | (a): omitted the chain rule factor |
| Q5 | (b) 6 | (c): gave the $y$-value instead of the slope |

Worked note for Q2:

$$
\int_0^3 2x\,dx = \left[x^2\right]_0^3 = 9 - 0 = 9
$$

### 6.2 Section B

**Q6.** Product rule $(uv)' = u'v + uv'$ with $u = x^2$, $v = \sin x$:

$$
f'(x) = 2x\sin x + x^2\cos x
$$

Marks: rule stated or applied correctly [2]; each term correct [1 + 1].

**Q7.** Differentiate and set to zero:

$$
f'(x) = 3x^2 - 3 = 3(x - 1)(x + 1) = 0 \implies x = -1 \text{ or } x = 1
$$

Second derivative $f''(x) = 6x$. At $x = 1$, $f''(1) = 6 > 0$: local minimum,
$f(1) = -2$. At $x = -1$, $f''(-1) = -6 < 0$: local maximum, $f(-1) = 2$.

Marks: $f'(x)$ [1]; both critical points [1]; correct classification of each
with justification [1 + 1]. Accept a sign chart of $f'$ instead of $f''$.

**Q8.** Divide by 2 and factor:

$$
x^2 - 4x + 3 = (x - 1)(x - 3) = 0 \implies x = 1 \text{ or } x = 3
$$

Marks: simplification or correct use of the quadratic formula [2]; each root
[1 + 1]. Check: $2(1)^2 - 8(1) + 6 = 0$ and $2(9) - 24 + 6 = 0$.

### 6.3 Section C

**Q9 (a).** Chain rule with outer function $u^4$ and inner $u = 3x^2 + 1$:

$$
f'(x) = 4\left(3x^2 + 1\right)^3 \cdot 6x = 24x\left(3x^2 + 1\right)^3
$$

Marks: derivative of outer function [2]; derivative of inner function [2];
correct product and simplification [2].

**Q9 (b).**

$$
f'(1) = 24(1)\left(3 + 1\right)^3 = 24 \times 64 = 1536
$$

Marks: substitution [1]; arithmetic [2]. Follow-through: award full marks for
correct evaluation of an incorrect $f'(x)$ from part (a).

**Q9 (c).** $f(0) = (0 + 1)^4 = 1$ and $f'(0) = 0$, so the tangent line is

$$
y - 1 = 0\,(x - 0) \implies y = 1
$$

Marks: point $f(0)$ [1]; slope $f'(0)$ [2]; line equation [2].

**Q10 (a).** Intersections: $x^2 = 2x \implies x(x - 2) = 0 \implies x = 0, 2$.
On $[0, 2]$, $2x \ge x^2$, so

$$
A = \int_0^2 \left(2x - x^2\right) dx = \left[x^2 - \frac{x^3}{3}\right]_0^2 = 4 - \frac{8}{3} = \frac{4}{3}
$$

Marks: limits of integration [2]; correct integrand with the upper curve
first [1]; antiderivative [3]; final value [2]. Accept $\approx 1.33$.

**Q10 (b).** Adding the equations eliminates $y$:

$$
3x = 9 \implies x = 3, \qquad y = 7 - 2(3) = 1
$$

Marks: valid elimination or substitution [3]; $x$ [1]; $y$ [1]; check in both
equations [1].

## 7. Marking guidance

<!-- Guidance: Shared rules make marking consistent across markers and fair to students. Record decisions on unexpected answers so they are applied to every script. -->

- Award method marks for correct reasoning even when an arithmetic slip leads
  to a wrong final answer.
- Apply follow-through: an error in one part is penalised once, and later
  parts are marked on the student's own earlier value.
- Accept any correct equivalent form (for example $2x\sin x + x^2\cos x$ or
  $x(2\sin x + x\cos x)$).
- Do not award marks for a correct answer obtained by clearly invalid working.
- Use half-marks only where the scheme shows them.
- Second-mark a sample of [10%] of scripts, including all scripts within [2]
  points of a grade boundary.

| Date | Question | Unexpected answer | Decision |
|---|---|---|---|
| *Example:* YYYY-MM-DD | Q7 | Classified points using a graph sketch only | 1 of 2 classification marks: justification incomplete |
| [Date] | [Question] | [Answer] | [Decision] |

### Results summary

| Statistic | Value |
|---|---|
| Scripts marked | [n] |
| Mean (SD) | [ ] |
| Questions with mean below 50% | [ ] |
| Action for next exam or teaching | [ ] |

## Standard and licence note

Based on common assessment practice for written examinations and marking
schemes. Original text written for this collection; sample questions and
solutions are illustrative and were checked for correctness. Template
licence: MIT.