Elementary Mathematics II — 2025A Past Questions

University of Ilorin MTH102: 70 questions.

Topics: Derivatives as limits of rate of change, differentiation techniques, maxima and minima., Functions of a real variable, graphs, limits, and continuity.

1. Find the derivative of $y = x e^{-x^2}$.
   1. $e^{-x^2}(1 - 2x^2)$
   2. $2x e^{-x^2}$
   3. $-2x e^{-x^2}$
   4. $e^{-x^2}(2x^2 - 1)$
2. If $y = \cos 2t$ and $x = 2t + 5$, find $dy/dx$.
   1. $-\sin 2t$
   2. $-2\sin 2t$
   3. $\cos 2t$
   4. $2\cos 2t$
3. If $y = \ln(x + x^7)$, find $dy/dx$ at $x = 1$.
   1. $4$
   2. $8$
   3. $1/2$
   4. $2$
4. Differentiate $y = \ln(3x + 2)^5$ with respect to $x$.
   1. $\frac{15}{3x+2}$
   2. $\frac{5}{3x+2}$
   3. $\frac{3}{3x+2}$
   4. $\frac{15}{(3x+2)^5}$
5. Find the second derivative $\frac{d^2y}{dx^2}$ of $y = \tan^{-1}x$.
   1. $\frac{-2x}{(1+x^2)^2}$
   2. $\frac{2x}{(1+x^2)^2}$
   3. $\frac{-1}{(1+x^2)^2}$
   4. $\frac{1}{(1+x^2)}$
6. Find the fifth derivative of $y = e^{-2x} + x^4$.
   1. $-32e^{-2x}$
   2. $32e^{-2x}$
   3. $-16e^{-2x} + 24$
   4. $-64e^{-2x}$
7. Let $f(x) = x^2 - 3x + 2$. Find $f(3) / f(4)$.
   1. $1/3$
   2. $3$
   3. $1/2$
   4. $2/3$
8. Suppose $\lim_{x \to a} f(x) = 86$ and $\lim_{x \to a} g(x) = 15$. Find $\lim_{x \to a} [f(x) - 2g(x)]$.
   1. $56$
   2. $71$
   3. $116$
   4. $101$
9. Suppose $f(x) = 9x^2 + 5$. Find the value of $f(-2)$.
   1. $41$
   2. $-31$
   3. $31$
   4. $23$
10. Find the derivative of $y = 3^x$.
    1. $3^x \ln 3$
    2. $x 3^{x-1}$
    3. $3^x \log_3 e$
    4. $\ln 3$
11. Find the derivative of $y = 4x^3 \sin x$.
    1. $4x^2(3\sin x + x\cos x)$
    2. $12x^2 \cos x$
    3. $12x^2 \sin x - 4x^3 \cos x$
    4. $4x^2(3\cos x + x\sin x)$
12. What is the derivative of $y = \sin 3x$?
    1. $3\cos 3x$
    2. $\cos 3x$
    3. $-3\cos 3x$
    4. $3\sin 3x$
13. Find $d/dx [x^3 \ln(x^2 - 2x + 5)]$.
    1. $x^2[3\ln(x^2 - 2x + 5) + \frac{x(2x - 2)}{x^2 - 2x + 5}]$
    2. $x^2[3\ln(x^2 - 2x + 5) - \frac{x(2x - 2)}{x^2 - 2x + 5}]$
    3. $\frac{3x^2}{x^2 - 2x + 5}$
    4. $3x^2 \ln(x^2 - 2x + 5)$
14. Find the derivative of $y = \sin^2(2x + 5)$ with respect to $x$.
    1. $4\sin(2x + 5)\cos(2x + 5)$
    2. $2\cos(2x + 5)$
    3. $-4\sin(2x + 5)\cos(2x + 5)$
    4. $4\cos^2(2x + 5)$
15. Find the second derivative of $y = x^x$.
    1. $x^x[(\ln x + 1)^2 + 1/x]$
    2. $x^x(\ln x + 1)$
    3. $x^x[(\ln x + 1)^2 - 1/x]$
    4. $x^{x-1}$
16. Given that $g(x) = x^3 + 2x^2$, find $g(3) / g(2)$.
    1. $45/16$
    2. $15/8$
    3. $27/8$
    4. $45/8$
17. If $\lim_{x \to a} f(x) = 36$ and $\lim_{x \to a} g(x) = 15$, find $\lim_{x \to a} [f(x) - 2g(x)]$.
    1. $6$
    2. $16$
    3. $26$
    4. $36$
18. Find the derivative of $y = \ln(5x)$.
    1. $1/x$
    2. $5/x$
    3. $1/(5x)$
    4. $5\ln(5x)$
19. Find the derivative of $y = \frac{4e^x}{\sin x}$.
    1. $\frac{4e^x(\sin x - \cos x)}{\sin^2 x}$
    2. $\frac{4e^x(\cos x - \sin x)}{\sin^2 x}$
    3. $\frac{4e^x(\sin x + \cos x)}{\sin^2 x}$
    4. $\frac{4e^x}{\cos x}$
20. If $y = 3 + 2t^2$ and $x = 2 - 3t^2$, find $dy/dx$.
    1. $-2/3$
    2. $3/2$
    3. $-4/6$
    4. $2/3$
21. If $y = x^{x^2}$, find $dy/dx$.
    1. $x^{x^2}[2x\ln x + x]$
    2. $x^{x^2}[x\ln x + 2x]$
    3. $x^{x^2}[2x\ln x - x]$
    4. $x^{2} x^{x^2-1}$
22. Find $dy/dx$ given that $x^2 + y^2 - 3xy = -1$ at $y=1$ and $x=1$.
    1. $-1$
    2. $1$
    3. $0$
    4. $2$
23. Find the first derivative of $x = f(t)$.
    1. $\lim_{\delta t \to 0} \frac{\delta x}{\delta t}$
    2. $\lim_{\delta t \to \infty} \frac{\delta x}{\delta t}$
    3. $\lim_{\delta x \to 0} \frac{\delta t}{\delta x}$
    4. $\lim_{\delta t \to 0} \frac{\delta y}{\delta t}$
24. Differentiate $x^2 - xy + y^2 = 7$ and find $dy/dx$ taking $x=3$ and $y=2$.
    1. $-4$
    2. $4$
    3. $-2$
    4. $1$
25. Given $f(x) = x^2 - x - 2$, find $f(2a) - f(a)$.
    1. $3a^2 - a$
    2. $a^2 - a$
    3. $3a^2 + a$
    4. $2a^2 - a$
26. Differentiate $y = \sin^3 x$.
    1. $3\sin^2 x \cos x$
    2. $\cos^3 x$
    3. $3\cos^2 x \sin x$
    4. $-3\sin^2 x \cos x$
27. Find the maximum point on the curve $y = \frac{1}{3}x^3 - \frac{1}{2}x^2 - 2x + 4$.
    1. $(-1, 31/6)$
    2. $(2, 2/3)$
    3. $(1, 31/6)$
    4. $(2, 31/6)$
28. Find $D_x [\sin^2(x^2 + 1)]$.
    1. $2x\sin(2x^2 + 2)$
    2. $4x\sin(x^2 + 1)\cos(x^2 + 1)$
    3. $2x\cos(2x^2 + 2)$
    4. $4x\cos^2(x^2 + 1)$
29. Calculate $\lim_{x \to 5} \frac{x^2 - 25}{x - 5}$.
    1. $10$
    2. $0$
    3. $5$
    4. Undefined
30. If $f(x) = 6x^2 - 7x$, find $f(-x)$.
    1. $6x^2 + 7x$
    2. $-6x^2 + 7x$
    3. $6x^2 - 7x$
    4. $-6x^2 - 7x$
31. Differentiate $y = \log_a (x^2 - 1)$.
    1. $\frac{2x}{(x^2 - 1)\ln a}$
    2. $\frac{2x}{x^2 - 1}$
    3. $\frac{x}{(x^2 - 1)\ln a}$
    4. $\frac{2x\ln a}{x^2 - 1}$
32. Find the derivative of $y = e^{e^x}$.
    1. $e^{e^x} e^x$
    2. $e^{e^x}$
    3. $e^x e^{e^x-1}$
    4. $x e^{e^x}$
33. Find the fourth derivative of $y = 3^x$.
    1. $3^x(\ln 3)^4$
    2. $3^x\ln 3$
    3. $3^x(\ln 3)^3$
    4. $x^4 3^{x-4}$
34. What is the derivative of $y = \ln x$?
    1. $1/x$
    2. $x$
    3. $-1/x^2$
    4. $e^x$
35. Evaluate $\lim_{x \to 2} (3x^2 + 4x + 6)$.
    1. $26$
    2. $24$
    3. $16$
    4. $30$
36. Derivative of $y = \ln(\sqrt{1/x})$ is?
    1. $-1/(2x)$
    2. $1/(2x)$
    3. $-1/x$
    4. $-2/x$
37. Calculate $\lim_{x \to 4} \frac{x^3 - 64}{x - 4}$.
    1. $48$
    2. $16$
    3. $0$
    4. Undefined
38. Find $dy/dx$ if $y = \frac{2t}{1+t}$ and $x = \frac{1-t}{1+t}$.
    1. $-1$
    2. $1$
    3. $-2$
    4. $2/(1+t)^2$
39. Evaluate $\lim_{x \to 5} \frac{x^2 + 1}{x - 1}$.
    1. $6.5$
    2. $13$
    3. $26$
    4. $13/3$
40. Derivative of $y = \frac{1}{1+x^2}$ is?
    1. $\frac{-2x}{(1+x^2)^2}$
    2. $\frac{2x}{(1+x^2)^2}$
    3. $\frac{1}{(1+x^2)^2}$
    4. $\frac{-1}{(1+x^2)^2}$
41. Derivative of $y = \ln(2x+1)$ is?
    1. $\frac{2}{2x+1}$
    2. $\frac{1}{2x+1}$
    3. $\frac{2x}{2x+1}$
    4. $\frac{1}{(2x+1)^2}$
42. What is the derivative of $y = x e^x \ln 5$?
    1. $e^x \ln 5(x + 1)$
    2. $e^x \ln 5(x - 1)$
    3. $x e^x \ln 5$
    4. $e^x \ln 5$
43. Find the derivative of $y = e^x \ln(x^2)$.
    1. $2e^x(\ln x + 1/x)$
    2. $e^x(\ln x^2 + 1/x)$
    3. $2e^x \ln x$
    4. $e^x \ln x^2$
44. Find the derivative of $y = \tan x$.
    1. $\sec^2 x$
    2. $\cot^2 x$
    3. $\sin^2 x$
    4. $\cos^2 x$
45. If $y = \frac{\tan x}{e^x}$, find $dy/dx$.
    1. $\frac{\sec^2 x - \tan x}{e^x}$
    2. $\frac{\tan x - \sec^2 x}{e^x}$
    3. $\frac{\sec^2 x + \tan x}{e^x}$
    4. $\frac{\sec^2 x}{e^x}$
46. If $y = \tan^{-1}(\ln x)$, find $dy/dx$.
    1. $\frac{1}{x[1 + (\ln x)^2]}$
    2. $\frac{1}{1 + (\ln x)^2}$
    3. $\frac{x}{1 + (\ln x)^2}$
    4. $\frac{\ln x}{1 + x^2}$
47. If $y = \ln x$, find $d^5y/dx^5$.
    1. $24/x^5$
    2. $-24/x^5$
    3. $6/x^4$
    4. $-6/x^4$
48. Find $dy/dx$ if $y = \frac{\cos x}{x \ln 2}$.
    1. $\frac{-(x\sin x + \cos x)}{x^2 \ln 2}$
    2. $\frac{x\sin x + \cos x}{x^2 \ln 2}$
    3. $\frac{-(x\sin x - \cos x)}{x^2 \ln 2}$
    4. $\frac{-\sin x}{x \ln 2}$
49. Find the fifth derivative of $y = -2e^{-2x} + 4x^3$.
    1. $64e^{-2x}$
    2. $-64e^{-2x}$
    3. $32e^{-2x}$
    4. $-32e^{-2x}$
50. If $x = \cos t$ and $y = \sin t$, find $dy/dx$.
    1. $-\cot t$
    2. $-\tan t$
    3. $\cot t$
    4. $\tan t$
51. Differentiate $y = \cos^{-1} 3x$.
    1. $\frac{-3}{\sqrt{1 - 9x^2}}$
    2. $\frac{3}{\sqrt{1 - 9x^2}}$
    3. $\frac{-1}{\sqrt{1 - 9x^2}}$
    4. $\frac{-3}{\sqrt{1 - 3x^2}}$
52. Given that $x^2 + y^2 - 2x + 2y = 23$, find $dy/dx$ at points $x=-2, y=3$.
    1. $3/4$
    2. $-3/4$
    3. $4/3$
    4. $-4/3$
53. If $y = \ln(x^3)$, find $dy/dx$.
    1. $3/x$
    2. $1/x^3$
    3. $3x^2$
    4. $1/x$
54. If $f(x) = (x+1)^2$, evaluate $f(1) + f(6)$.
    1. $53$
    2. $49$
    3. $4$
    4. $54$
55. If $y = x^x$ and $\ln 2 = 0.693$, find $dy/dx$ at $x=2$.
    1. $6.772$
    2. $4.000$
    3. $2.772$
    4. $8.000$
56. Find the derivative of $y = 3^{x^x}$.
    1. $3^{x^x} \ln 3 \cdot x^x(1 + \ln x)$
    2. $3^{x^x} \ln 3$
    3. $x^x(1 + \ln x)$
    4. $3^{x^x} x^x$
57. The minimum turning point of the curve $y = 2x^2 - 8x + 3$ is at what point?
    1. $(2, -5)$
    2. $(2, 5)$
    3. $(-2, 5)$
    4. $(-2, -5)$
58. Find $dy/dx$ given that $x = t/(1+t)$ and $y = 1/(1+t)$.
    1. $-1$
    2. $1$
    3. $-t$
    4. $t$
59. Derivative of $y = \sin^{-1}(2-x^2)$ is?
    1. $\frac{-2x}{\sqrt{4x^2 - x^4 - 3}}$
    2. $\frac{2x}{\sqrt{4x^2 - x^4 - 3}}$
    3. $\frac{-2x}{\sqrt{1 - (2-x^2)^2}}$
    4. $\frac{-x}{\sqrt{4x^2 - x^4 - 3}}$
60. If $f(x) = x^2 - 3x + 6$, what is $f(5) - f(3)$?
    1. $10$
    2. $16$
    3. $6$
    4. $0$
61. Derivative of $y = \frac{3}{(1-2x)^2}$ is?
    1. $\frac{12}{(1-2x)^3}$
    2. $\frac{-12}{(1-2x)^3}$
    3. $\frac{6}{(1-2x)^3}$
    4. $\frac{-6}{(1-2x)^3}$
62. What is the derivative of $y = \sin^2(3x + \pi/6)$?
    1. $3\sin(6x + \pi/3)$
    2. $6\sin(3x + \pi/6)\cos(3x + \pi/6)$
    3. $3\cos^2(3x + \pi/6)$
    4. $\sin(6x + \pi/3)$
63. Evaluate $d/dx [(2\ln x)^2]$.
    1. $8\ln x / x$
    2. $4\ln x / x$
    3. $8/x^2$
    4. $4/x^2$
64. Find the third derivative of $y = \sin^{-1}[\cos(2x^4+1)]$.
    1. $-48x$
    2. $-24x^2$
    3. $-8x^3$
    4. $48x$
65. Differentiate $y=x^2$ from first principles.
    1. $2x$
    2. $x^2$
    3. $2$
    4. $2x + \delta x$
66. Differentiate $y = \ln x$ from first principles.
    1. $1/x$
    2. $\ln x / x$
    3. $1/x^2$
    4. $x$
67. Find the derivative of $y = 2^x e^x$.
    1. $2^x e^x(\ln 2 + 1)$
    2. $2^x e^x \ln 2$
    3. $2^x e^x(x + 1)$
    4. $2e^{x+1}$
68. Derivative of $y = \frac{x^2+1}{x-1}$ is?
    1. $\frac{x^2 - 2x - 1}{(x-1)^2}$
    2. $\frac{x^2 + 2x - 1}{(x-1)^2}$
    3. $\frac{x^2 - 2x + 1}{(x-1)^2}$
    4. $\frac{2x}{(x-1)^2}$
69. Evaluate the limit $\lim_{\delta x \to 0} \frac{2 \cos(x) \sin(\delta x / 2)}{\delta x / 2}$ which is related to differentiating $\sin x$ from first principles.
    1. $2\cos x$
    2. $\cos x$
    3. $\sin x$
    4. $2\sin x$
70. Derivative of $y = \ln(\sin 3x)$ is?
    1. $3\cot 3x$
    2. $3\tan 3x$
    3. $\cot 3x$
    4. $3\cos 3x / \sin x$

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