Level 1 — RecognitionCalculus I — Limits & Derivatives

Calculus I — Limits & Derivatives

⏱ 20 minutes30 marksprintable — key stays hidden on paper

Level 1 — Recognition Test

Time limit: 20 minutes Total marks: 30

Section A — Multiple Choice (1 mark each)

Choose the single best answer.

Q1. $\displaystyle\lim_{x\to 0}\frac{\sin x}{x}$ equals: (a) $0$ (b) $1$ (c) $\infty$ (d) undefined

Q2. The derivative of $x^n$ (power rule) is: (a) $nx^{n-1}$ (b) $nx^{n+1}$ (c) $\frac{x^{n+1}}{n+1}$ (d) $(n-1)x^n$

Q3. $\displaystyle\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^{x}$ equals: (a) $1$ (b) $0$ (c) $e$ (d) $\infty$

Q4. The derivative of $\sin x$ is: (a) $\cos x$ (b) $-\cos x$ (c) $-\sin x$ (d) $\sec^2 x$

Q5. A function $f$ is continuous at $x=a$ if: (a) $f(a)$ exists only (b) $\lim_{x\to a}f(x)$ exists only (c) $\lim_{x\to a}f(x)=f(a)$ (d) $f$ is differentiable at $a$

Q6. The derivative of $e^x$ is: (a) $xe^{x-1}$ (b) $e^x$ (c) $\frac{e^x}{x}$ (d) $\ln x$

Q7. For $f(x)=\dfrac{1}{x}$ , the line $x=0$ is a: (a) horizontal asymptote (b) vertical asymptote (c) removable discontinuity (d) inflection point

Q8. The derivative of $\ln x$ is: (a) $\frac{1}{x}$ (b) $x$ (c) $e^x$ (d) $\frac{1}{x\ln 10}$

Q9. The product rule states $(uv)'=$ (a) $u'v'$ (b) $u'v+uv'$ (c) $u'v-uv'$ (d) $\frac{u'v-uv'}{v^2}$

Q10. The difference quotient defining $f'(x)$ is: (a) $\dfrac{f(x+h)-f(x)}{h}$ (b) $\dfrac{f(x)-f(a)}{x}$ (c) $f(x+h)-f(x)$ (d) $\dfrac{f(x)}{h}$

Q11. The Intermediate Value Theorem requires the function to be: (a) differentiable (b) continuous on a closed interval (c) increasing (d) bounded

Q12. A jump discontinuity occurs when: (a) both one-sided limits exist but are unequal (b) the limit equals the value (c) the function is undefined everywhere (d) the derivative is infinite

Section B — Matching (1 mark each; total 6)

Q13. Match each function to its derivative:

Function		Derivative
(i) $\cos x$		(A) $\sec^2 x$
(ii) $\tan x$		(B) $-\sin x$
(iii) $a^x$		(C) $\frac{1}{\sqrt{1-x^2}}$
(iv) $\arcsin x$		(D) $a^x\ln a$

(4 marks — one per correct pair)

Q14. Match the theorem/rule to its statement (2 marks):

Item		Statement
(i) Rolle's Theorem		(P) $\lim\frac{f}{g}=\lim\frac{f'}{g'}$ for $0/0$
(ii) L'Hôpital's Rule		(Q) If $f(a)=f(b)$ , some $c$ has $f'(c)=0$

Section C — True/False WITH Justification (2 marks each; 1 T/F + 1 reason)

Q15. The Squeeze Theorem can be used to show $\displaystyle\lim_{x\to0}x^2\sin\frac1x=0$ . True/False — justify.

Q16. If $\lim_{x\to a}f(x)$ exists, then $f$ is continuous at $a$ . True/False — justify.

Q17. The quotient rule gives $\left(\dfrac{u}{v}\right)'=\dfrac{u'v-uv'}{v^2}$ . True/False — justify.

Q18. At a local maximum of a differentiable function, $f'(x)=0$ . True/False — justify.

Answer keyMark scheme & solutions

Section A (12 marks)

Q1. (b) $1$ — the fundamental trig limit. (1)

Q2. (a) $nx^{n-1}$ — power rule. (1)

Q3. (c) $e$ — definition of $e$ . (1)

Q4. (a) $\cos x$ — standard derivative. (1)

Q5. (c) $\lim_{x\to a}f(x)=f(a)$ — all three conditions (limit exists, value exists, equal) summarised by this equation. (1)

Q6. (b) $e^x$ — $e^x$ is its own derivative. (1)

Q7. (b) vertical asymptote — as $x\to0$ , $1/x\to\pm\infty$ . (1)

Q8. (a) $\frac1x$ — derivative of natural log. (1)

Q9. (b) $u'v+uv'$ — product rule. (1)

Q10. (a) $\dfrac{f(x+h)-f(x)}{h}$ — first-principles difference quotient (limit as $h\to0$ ). (1)

Q11. (b) continuous on a closed interval — IVT hypothesis. (1)

Q12. (a) both one-sided limits exist but are unequal — definition of jump discontinuity. (1)

Section B (6 marks)

Q13. (i)→(B), (ii)→(A), (iii)→(D), (iv)→(C). (1 each = 4)

$\frac{d}{dx}\cos x=-\sin x$ ; $\frac{d}{dx}\tan x=\sec^2x$ ; $\frac{d}{dx}a^x=a^x\ln a$ ; $\frac{d}{dx}\arcsin x=\frac{1}{\sqrt{1-x^2}}$ .

Q14. (i)→(Q), (ii)→(P). (1 each = 2)

Section C (8 marks)

Q15. True. (1) Since $-1\le\sin\frac1x\le1$ , we get $-x^2\le x^2\sin\frac1x\le x^2$ . Both bounds $\to0$ as $x\to0$ , so by the Squeeze Theorem the middle $\to0$ . (1)

Q16. False. (1) Existence of the limit is not enough — we also need $f(a)$ defined and equal to the limit. E.g. a removable discontinuity has a limit but $f(a)$ may differ or be undefined. (1)

Q17. True. (1) This is the correct quotient rule (numerator $u'v-uv'$ , denominator $v^2$ ), valid where $v\neq0$ . (1)

Q18. True. (1) By Fermat's theorem, at an interior local extremum of a differentiable function the tangent is horizontal, so $f'(x)=0$ (a necessary condition). (1)

[
  {"claim":"limit sin(x)/x -> 1","code":"x=symbols('x'); result = (limit(sin(x)/x, x, 0)==1)"},
  {"claim":"limit (1+1/x)**x -> e","code":"x=symbols('x'); result = (limit((1+1/x)**x, x, oo)==E)"},
  {"claim":"derivative of tan(x) is sec(x)**2","code":"x=symbols('x'); result = simplify(diff(tan(x),x)-sec(x)**2)==0"},
  {"claim":"squeeze: limit x**2 sin(1/x) -> 0","code":"x=symbols('x'); result = (limit(x**2*sin(1/x), x, 0)==0)"}
]