Applications of Differential Calculus — Book-Back Answer Key (Exercise 7.10)

Exercise 7.10 — Book-Back Answer Key

Correct option for each multiple-choice question, with a one-line reason. Every answer below was independently re-derived; all 20 agree with the official key.

1. Q1. Option (1) [A] — \( \frac{dV}{dt}=4\pi r^2\frac{dr}{dt}\Rightarrow 3\pi=\pi\frac{dr}{dt}\Rightarrow\frac{dr}{dt}=3 \) cm/s.
2. Q2. Option (2) [B] — \( \tan\theta=\frac{y}{40} \); at \( y=30 \), \( \sec^2\theta=\frac{25}{16} \), giving \( \frac{d\theta}{dt}=\frac{4}{25} \).
3. Q3. Option (2) [B] — At rest \( \Rightarrow v=s'(t)=6t-2=0\Rightarrow t=\frac{1}{3} \).
4. Q4. Option (2) [B] — Max height when \( \frac{dx}{dt}=80-32t=0\Rightarrow t=2.5 \) s.
5. Q5. Option (1) [A] — \( \frac{dy}{dt}=8\frac{dx}{dt}\Rightarrow 48=3x^2\Rightarrow x=4,\;y=11 \).
6. Q6. Option (2) [B] — \( f'(x)=\frac{-1}{\sqrt{8-2x}}=-\frac14\Rightarrow 8-2x=16\Rightarrow x=-4 \).
7. Q7. Option (3) [C] — Tangent slope \( =-8\sin\frac{\pi}{3}=-4\sqrt3 \); normal slope \( =\frac{1}{4\sqrt3}=\frac{\sqrt3}{12} \).
8. Q8. Option (4) [D] — \( \frac{dy}{dx}=\frac{y}{2y-x} \) is undefined when \( x=2y \); substituting gives \( y^2=9\Rightarrow y=\pm3 \).
9. Q9. Option (3) [C] — At the origin the tangents are the \( y \)-axis and the \( x \)-axis, which meet at \( \frac{\pi}{2} \).
10. Q10. Option (1) [A] — Combine to \( \frac{x\cos x-\sin x}{x\sin x} \) (form \( \frac00 \)); two L'Hopital steps give \( 0 \).
11. Q11. Option (3) [C] — \( f=\frac34+\frac14\cos4x\Rightarrow f'=-\sin4x>0 \) on \( \left[\frac{\pi}{4},\frac{\pi}{2}\right] \) where \( 4x\in[\pi,2\pi] \).
12. Q12. Option (4) [D] — \( f(0)=f(3)=0 \); \( f'(x)=3x(x-2)=0 \) gives \( c=2 \) in \( (0,3) \).
13. Q13. Option (3) [C] — \( f'(c)=\frac{f(9)-f(1)}{8}=-\frac19 \) and \( f'(x)=-\frac{1}{x^2}\Rightarrow c=3 \).
14. Q14. Option (4) [D] — \( |3-x|\ge0 \) with minimum \( 0 \) at \( x=3 \), so the least value is \( 9 \).
15. Q15. Option (2) [B] — Slope \( m=e^x(\sin x+\cos x) \), \( m'=2e^x\cos x=0\Rightarrow x=\frac{\pi}{2} \) gives the largest slope.
16. Q16. Option (3) [C] — \( f'(x)=2x(1-x)e^{-2x}=0\Rightarrow x=1 \) (max), \( f(1)=\frac{1}{e^{2}} \).
17. Q17. Option (3) [C] — Minimise \( D=2x^2-12x+32\Rightarrow x=3,\;y^2=5 \), giving \( (3,\sqrt5) \).
18. Q18. Option (1) [A] — Maximise \( P^2=x^2(200-x^2)\Rightarrow x^2=100=y^2 \), so \( P=10\cdot10=100 \).
19. Q19. Option (4) [D] — \( y''=12ax^2+2b=0\Rightarrow x^2=-\frac{b}{6a}<0 \) (since \( ab>0 \)): no real inflection point.
20. Q20. Option (3) [C] — \( y''=6(x-1)=0 \) at \( x=1 \) with a sign change; there \( y=0 \), so \( (1,0) \).