📝 Unit 1: Introduction to Calculus — Unit Test

Assessment OF Learning — Summative
✅ Graded — Counts Toward 70% Term Mark
⏱️ Duration: 75 minutes  |  Total: /60 marks
Show all work. No work, no marks.
K/U
/15
Thinking
/15
Comm.
/15
Applic.
/15
Part A: Knowledge & Understanding [15 marks]
1 [2]
Evaluate \( \displaystyle \lim_{x\to 4}\frac{x^2-16}{x-4} \).
2 [3]
Evaluate \( \displaystyle \lim_{x\to 0}\frac{\sqrt{x+4}-2}{x} \).
3 [2]
MC: \( \displaystyle \lim_{x\to\infty}\frac{4x^3-x}{2x^3+5} = \)
4 [3]
Find the value of \(k\) so that \(f(x)=\begin{cases}\frac{x^2-9}{x-3},& x\ne 3\\ k,& x=3\end{cases}\) is continuous at \(x=3\).
5 [5]
Use first principles to determine \(f'(x)\) for \(f(x)=2x^2-5x+1\). Then state \(f'(3)\).
Part B: Thinking [15 marks]
6 [5]
Use first principles to find \(f'(x)\) for \(f(x)=\frac{1}{x+2}\). Verify your result agrees with the power-rule answer (you may state the rule answer).
7 [5]
For \(f(x)=\sqrt{x+1}\), use first principles to determine the slope of the tangent at \(x=3\).
8 [5]
A function \(g\) has \(\lim_{x\to 1^-} g(x)=2\), \(\lim_{x\to 1^+} g(x)=5\), and \(g(1)=2\). Determine which conditions of continuity are met and which fail. Suggest one way to redefine \(g\) at \(x=1\) so the (already-existing) limit at 1 from each side could be made consistent.
Part C: Communication [15 marks]
9 [4]
Explain in detail the formal definition \( f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h} \). Describe what each component represents geometrically.
10 [4]
Compare and contrast the three types of discontinuity (removable, jump, infinite). Sketch a description of each and give an example function.
11 [4]
A student writes "\(\lim_{x\to 2}\frac{x^2-4}{x-2}=\frac{0}{0}\), so the limit does not exist." Identify the error and write the correct argument.
12 [3]
Describe a procedure for evaluating \(\lim_{x\to a}f(x)\) when direct substitution gives \(0/0\). List at least three algebraic techniques.
Part D: Application [15 marks]
13 [5]
A stone falls from a 80 m cliff: \(s(t)=80-5t^2\) (m, s).
a) Find the average velocity on \([1,3]\).
b) Find the instantaneous velocity at \(t=2\) using first principles.
c) When does the stone hit the ground?
14 [5]
A water tank is being drained: \(V(t)=200(1-t/20)^2\) litres for \(0\le t\le 20\) min.
a) State \(V(0)\) and \(V(20)\) and interpret.
b) Find the average rate of change of \(V\) on \([0,10]\).
c) Estimate the instantaneous rate of change at \(t=10\) by computing the difference quotient with \(h=0.01\).
15 [5]
A drug concentration model: \(C(t)=\frac{6t}{t^2+4}\) (mg/L, hours).
a) Find \(\lim_{t\to\infty}C(t)\) and interpret.
b) Determine \(C(2)\) (the candidate maximum).
c) Estimate the instantaneous rate of change at \(t=1\) using a small \(h\).

Evaluation Rubric

LevelDescription%
4Thorough, insightful80–100
3Considerable (provincial standard)70–79
2Some effectiveness60–69
1Limited effectiveness50–59
RInsufficient<50
📕 Answer Key (click to reveal)

1. \( \frac{(x-4)(x+4)}{x-4}=x+4\to 8\).

2. Rationalize: \(\frac{1}{\sqrt{x+4}+2}\to 1/4\).

3. Same degree → 4/2 = 2 → answer (b).

4. \(\lim_{x\to 3}\frac{(x-3)(x+3)}{x-3}=6\), so \(k=6\).

5. \(f'(x)=4x-5\); \(f'(3)=7\). Limit work: \(\frac{2(x+h)^2-5(x+h)+1-(2x^2-5x+1)}{h}=\frac{4xh+2h^2-5h}{h}=4x+2h-5\to 4x-5\).

6. \(f'(x)=-\frac{1}{(x+2)^2}\). Common denominator: \(\frac{(x+2)-(x+h+2)}{h(x+h+2)(x+2)}=\frac{-1}{(x+h+2)(x+2)}\to -1/(x+2)^2\). Power rule (\(f=(x+2)^{-1}\)): \(-(x+2)^{-2}\). ✓

7. \(\frac{\sqrt{4+h}-2}{h}\cdot\frac{\sqrt{4+h}+2}{\sqrt{4+h}+2}=\frac{1}{\sqrt{4+h}+2}\to 1/4\).

8. Limit at 1 doesn't exist (left ≠ right). Continuity fails because 2-sided limit doesn't exist; redefine \(g(1)\) doesn't fix that. To make it continuous, you must change the definitions so that the one-sided limits agree — e.g. redefine the right-side rule.

9. \(f(x+h)-f(x)\) = rise; \(h\) = run; ratio = secant slope; limit \(h\to 0\) = tangent slope = instantaneous rate of change.

10. Removable: hole, e.g. \((x^2-1)/(x-1)\) at 1. Jump: piecewise with different one-sided limits. Infinite: vertical asymptote, e.g. \(1/x\) at 0.

11. Error: \(0/0\) is indeterminate, not "DNE." Correct: factor → \(x+2\to 4\).

12. (i) factor & cancel; (ii) rationalize numerator/denominator; (iii) common denominator (for complex fractions); (iv) divide by highest power (limits at infinity).

13. a) \(s(3)=35,s(1)=75\); ARoC = \(-20\) m/s. b) \(s'(t)=-10t\); \(s'(2)=-20\) m/s. c) \(0=80-5t^2\) → \(t=4\) s.

14. a) \(V(0)=200\) L (full); \(V(20)=0\) (empty). b) \(V(10)=50\); ARoC = \((50-200)/10=-15\) L/min. c) \(V'(t)=200\cdot 2(1-t/20)(-1/20)=-20(1-t/20)\); at 10: \(-10\) L/min (≈ matches \(h=0.01\) estimate).

15. a) Degree 1 over 2 → 0 mg/L (drug clears). b) \(C(2)=12/8=1.5\) mg/L. c) Quotient rule (or numerical): \(C'(t)=\frac{6(t^2+4)-6t(2t)}{(t^2+4)^2}=\frac{24-6t^2}{(t^2+4)^2}\); at 1: \(18/25=0.72\) mg/L/h.