📝 Unit 4: Curve Sketching — Unit Test

Extrema · Concavity · Asymptotes · Sketches
✅ Graded
⏱️ 75 min  |  Total: /60 marks
K/U
/15
Thinking
/15
Comm.
/15
Applic.
/15
Part A: K/U [15]
1 [3]
For \(f(x)=2x^3-3x^2-12x\), find all critical numbers and classify each (max/min) using the second-derivative test.
2 [3]
Determine intervals of concavity and any inflection points of \(g(x)=x^4-4x^3+6\).
3 [3]
State all asymptotes of \(h(x)=\dfrac{x^2-9}{x^2-4}\).
4 [3]
MC: For \(p(x)=x e^{-x}\), the first derivative is:
5 [3]
A graph has \(f'(x)=(x-1)^2(x+3)\). Determine all extrema. (Hint: even multiplicity at 1.)
Part B: Thinking [15]
6 [7]
Perform a complete curve analysis of \(f(x)=\dfrac{x^2}{x-1}\): domain, asymptotes (V, H, slant), critical numbers, intervals inc./dec., extrema, concavity, inflection. Provide a labeled sketch.
7 [5]
A function has \(f''(x)=6x-12\). Find all inflection points and intervals of concavity. If \(f(2)=5\), find an antiderivative-based reasonable \(f'(2)\) given \(f'(0)=0\).
8 [3]
Sketch a function satisfying ALL: \(f(0)=2\), \(f'(x)>0\) on \((-\infty,2)\), \(f'(x)<0\) on \((2,\infty)\), \(f''(x)<0\) on all reals, horizontal asymptote \(y=0\) as \(x\to\infty\).
Part C: Communication [15]
9 [4]
Describe a step-by-step procedure (the "algorithm") for sketching a polynomial function. Cover at least: zeros, end behaviour, critical points, concavity, inflection.
10 [4]
Compare the first-derivative test and the second-derivative test for classifying critical numbers. Give one situation where the second-derivative test fails.
11 [4]
Explain how to identify each type of asymptote (vertical, horizontal, slant) for a rational function. Use \(f(x)=\dfrac{2x^2+1}{x-1}\) as your example.
12 [3]
A student says: "If \(f''(c)=0\), then \(c\) is an inflection point." Critique this claim with an example.
Part D: Application [15]
13 [5]
A drug concentration model: \(C(t)=\dfrac{20t}{t^2+4}\) (mg/L). Find the time of maximum concentration and the maximum value. Sketch the curve.
14 [5]
Profit \(P(x)=-x^3+30x^2-200x-100\). Find production level for maximum profit and analyze concavity (over what range is profit growing faster?).
15 [5]
Population: \(P(t)=\dfrac{1000}{1+9e^{-0.5t}}\). Find horizontal asymptotes; find the inflection point (point of maximum growth rate).

Evaluation Rubric

LevelDescription%
4Thorough80–100
3Considerable70–79
2Some60–69
1Limited50–59
RInsufficient<50
📕 Answer Key

1. \(f'=6x^2-6x-12=6(x-2)(x+1)=0\) → \(x=-1,2\). \(f''=12x-6\); \(f''(-1)=-18<0\) → max; \(f''(2)=18>0\) → min.

2. \(g''=12x^2-24x=12x(x-2)\); concave up \(x<0\) and \(x>2\); concave down \(0

3. Vertical: \(x=\pm 2\). Horizontal: \(y=1\) (same degree, ratio 1).

4. Product rule: \(p'=e^{-x}+x(-e^{-x})=(1-x)e^{-x}\) → (c).

5. \(f'\) zeros at 1 (even mult) and \(-3\). At \(-3\): sign + → − (decreasing) hmm let's recheck: factor signs: pick \(x=-4\): \((25)(-1)=-\) <0; \(x=0\): \((1)(3)=3>0\); \(x=2\): \((1)(5)=5>0\). So \(f'\) goes − to + at \(-3\) → local minimum at \(x=-3\). At \(x=1\): + to + → no extremum (just inflection of \(f\)).

6. Domain: \(x\ne 1\). VA: \(x=1\). Slant: long division: \(x+1+\frac{1}{x-1}\) → \(y=x+1\). \(f'=\frac{x^2-2x}{(x-1)^2}=\frac{x(x-2)}{(x-1)^2}=0\) at \(x=0,2\). Sign: \(f'>0\) on \((-\infty,0)\) and \((2,\infty)\); \(<0\) on \((0,1)\) and \((1,2)\). Local max at \(x=0\), \(f(0)=0\); local min at \(x=2\), \(f(2)=4\). \(f''=\frac{2}{(x-1)^3}\) → concave up \(x>1\), down \(x<1\). No inflection (asymptote breaks domain).

7. \(f''=0\) at \(x=2\) → inflection. Concave down \(x<2\), up \(x>2\). \(f'(x)=3x^2-12x+C_1\); \(f'(0)=0\) → \(C_1=0\); \(f'(2)=12-24=-12\).

8. Inverted bell-like curve, peaking at \((2,?)\), strictly concave down, descending to 0 on the right and rising from \(-\infty\) on left. Pass through \((0,2)\).

9. Find domain & intercepts; end behaviour; \(f'\) → critical points & inc/dec; \(f''\) → concavity & inflection; combine.

10. 1st test uses sign change of \(f'\) (always works if \(f'\) defined nearby). 2nd uses \(f''(c)\) sign (faster but inconclusive when \(f''(c)=0\), e.g. \(f(x)=x^4\) at 0).

11. VA: where denominator = 0 and numerator ≠ 0. HA: degree comparison. Slant: when num degree exceeds denom by exactly 1; do long division. For \(\frac{2x^2+1}{x-1}\): VA \(x=1\); slant from division: \(2x+2+\frac{3}{x-1}\), \(y=2x+2\); no HA.

12. False — \(f(x)=x^4\) has \(f''(0)=0\) but no inflection (still concave up). Need \(f''\) to change sign.

13. \(C'=\frac{20(t^2+4)-20t(2t)}{(t^2+4)^2}=\frac{80-20t^2}{(t^2+4)^2}=0\) → \(t=2\). \(C(2)=40/8=5\) mg/L. Long-term \(\to 0\).

14. \(P'=-3x^2+60x-200\); roots: \(x=\frac{60\pm\sqrt{3600-2400}}{6}=\frac{60\pm\sqrt{1200}}{6}\approx 4.23, 15.77\). Max at \(x\approx 15.77\). \(P''=-6x+60=0\) at \(x=10\) (inflection — fastest growth).

15. As \(t\to\infty\): \(P\to 1000\); as \(t\to-\infty\): \(P\to 0\). Inflection where \(P=500\): \(1+9e^{-0.5t}=2\) → \(e^{-0.5t}=1/9\) → \(t=2\ln 9\approx 4.39\).