Elementary Mathematics II — integral Past Questions
University of Ilorin MTH102: 40 questions.
Topics: Derivatives as limits of rate of change, differentiation techniques, maxima and minima., Curve sketching, integration, definite integrals, reduction formulae., Functions of a real variable, graphs, limits, and continuity.
- Evaluate the definite integral $\\int_{2}^{5} 6x^3 dx$.
- $912.5$
- $913.5$
- $914.5$
- $915.5$
- If $y = (3x^2 - \\cos x)^2$, find $\\frac{dy}{dx}$.
- $2(3x^2 - \\cos x)(6x + \\sin x)$
- $2(3x - \\cos x)(6x + \\sin x)$
- $2(3 - \\cos x)(6 + \\sin x)$
- $2(6x + \\sin x)(3x - \\sin x)$
- Evaluate $\\lim_{x \\to 0} \\frac{\\sin 2x}{4x}$.
- $0$
- $1/4$
- $1/2$
- $4$
- Find $\\frac{dy}{dx}$ for the polynomial $2x^4 - 7x^3 + 4x^2 + 3x - 4$.
- $8x^4 - 21x + 8x + 3$
- $8x^3 - 21x^2 + 8x + 3$
- $6x^2 - 21x^2 + 8x$
- $6x^3 - 21x + 8x + 3$
- Evaluate the indefinite integral $\\int 4 dt$.
- $4x + C$
- $4t + C$
- $4t^2 + C$
- $4x^2 + C$
- Differentiate $y = 3e^x \\sin x$.
- $3e^x \\cos x$
- $3e^x \\sin x + 3e^x \\cos x$
- $3e^x \\sin x$
- $3e^x \\cos x - 3e^x \\sin x$
- If $y = (4x - 3)^6$, find the second derivative $\\frac{d^2y}{dx^2}$.
- $24(4x - 3)^5$
- $144(4x - 3)^4$
- $120(4x - 3)^4$
- $480(4x - 3)^4$
- Find $\\frac{dy}{dx}$ of $y = \\cos 2x$.
- $-2\\sin 2x$
- $2\\sin 2x$
- $\\cot 2x$
- $2\\cos 2x$
- Evaluate the indefinite integral $\\int \\sin 8x dx$.
- $\\frac{1}{8}\\cos 8x + C$
- $-\\frac{1}{8}\\cos 8x + C$
- $\\frac{1}{8}\\csc 8x + C$
- $\\frac{1}{8}\\sin 8x + C$
- Evaluate the definite integral $\\int_{0}^{1} (3x^2 + 4x + 6) dx$.
- $9$
- $3$
- $6$
- $-3$
- If the displacement is given by $S(t) = t^2 + t - 6$, find the third derivative $\\frac{d^3S}{dt^3}$.
- $2$
- $2t + 7$
- $0$
- $-2$
- If $F(x) = 8x^2 + \\tan x$, find $\\frac{dy}{dx}$.
- $8x + \\tan^2 x$
- $8x + \\csc^2 x$
- $16x + \\sec^2 x$
- $8 + \\sec^2 x$
- If $y = t^2 \\cos t + \\sin 4t$, find $\\frac{dy}{dt}$.
- $2t \\cos t - t^2 \\sin t + 4 \\cos 4t$
- $2t \\cos t + t^2 \\sin t + 4 \\cos 4t$
- $2t \\cos t - t^2 \\sin t - 4 \\cos 4t$
- $2t \\sin t - t^2 \\cos t + 4 \\cos 4t$
- If $y = \\ln(\\cos 4x)$, find $\\frac{dy}{dx}$.
- $4 \\cos 4x$
- $4 \\cot 4x$
- $-4 \\tan 4x$
- $4 \\tan 4x$
- If displacement is given by the function $s = 2t + 4t^2$, find the velocity at $t = 2$ seconds.
- $16 \\text{ m/s}$
- $18 \\text{ m/s}$
- $14 \\text{ m/s}$
- $12 \\text{ m/s}$
- Find the area under the curve determined by the definite integral $\\int_{0}^{2} (-3x^2 + 4x + 6) dx$.
- $10$
- $12$
- $-6$
- $18$
- Evaluate $\\int_{1}^{4} 4x^2 dx$.
- $84$
- $83$
- $82$
- $81$
- Evaluate the integral $\\int_{1}^{2} (1 + 3x^2) dx$.
- $8$
- $18$
- $100$
- $10$
- Evaluate the limit $\\lim_{x \\to y} \\frac{x^2 - y^2}{x - y}$.
- $x + y$
- $x^2y$
- $2xy$
- $x - y$
- If $f(x) = \\frac{4x}{2x^2 - 1}$, find $\\frac{dy}{dx}$.
- $\\frac{-8x^2 - 4}{(2x^2 - 1)^2}$
- $\\frac{8x^2 - 4}{(2x^2 - 1)^2}$
- $\\frac{8x^2 + 4}{(2x^2 - 1)^2}$
- $\\frac{-8x^2 + 4}{(2x^2 - 1)^2}$
- If $y = 3e^x \\sin x$, find $\\frac{dy}{dx}$.
- $3e^x \\cos x$
- $3e^x \\sin x + 3e^x \\cos x$
- $3e^x \\sin x$
- $3e^x \\cos x - 3e^x \\sin x$
- If $y = x^4 + x^3$, find $\\frac{dy}{dx}$.
- $4x + 3x$
- $4x^3 + 3x^2$
- $4x^2 + 3x$
- $4x^3 + 3x$
- If $y = \\sin^{-1}(x^3)$, find $\\frac{dy}{dx}$.
- $\\frac{x^3}{\\sqrt{1-x^6}}$
- $\\cos^{-1}(x^3)$
- $\\frac{3x^2}{\\sqrt{1-x^6}}$
- $\\frac{3x}{\\sqrt{1-x^2}}$
- If $y = 3\\cos 2t$ and $x = 2\\sin t$, find $\\frac{dy}{dx}$.
- $6\\cos t$
- $-6\\sin t$
- $6\\cos t \\sin t$
- $-6\\cos t$
- Differentiate the expression $4 \\ln(5y)$ implicitly with respect to $x$.
- $\\frac{4}{y} \\frac{dy}{dx}$
- $4 \\frac{dy}{dx}$
- $\\frac{4}{5y} \\frac{dy}{dx}$
- None
- Find the 4th term of the Maclaurin series expansion of $e^{3x}$.
- $3x$
- $1x$
- $\\frac{27x^3}{3!}$
- $\\frac{x^3}{3!}$
- Find $\\frac{dy}{dx}$ of $y = (2x+1)(1+x)$.
- $4x - 3$
- $4x^2 - 3x$
- $4x^2 - 3x^2$
- $4x + 3$
- If $f(x) = (x+1)^2$, find $f(1) + f(a)$.
- $a + 2a + 5$
- $a^2 + 2a + 5$
- $a^2 - 2a + 5$
- $a + 2a^2 + 5$
- Find $\\frac{dy}{dx}$ if $y = e^x \\sec x$.
- $e^x \\sec x \\tan x$
- $e^x \\sec x (1 + \\tan x)$
- $e^x \\tan x (1 + \\sec x)$
- $e^x(1 + \\tan x)$
- Evaluate $\\lim_{x \\to 1} \\frac{x^2 - 2x + 1}{x - 1}$.
- $1$
- $0$
- $\\infty$
- $-1$
- Find $\\frac{dy}{dx}$ of $y = \\frac{\\cot x}{\\cos x}$.
- $\\sec x(\\cot x \\tan x - \\csc^2 x)$
- $\\sec^2 x(\\cot x \\tan x + \\csc^2 x)$
- $\\sec x(\\cot^2 x \\tan x - \\csc^2 x)$
- $\\sec x(\\cot x \\tan x + \\csc^2 x)$
- If $y = 5 - 3t^2$, find $\\frac{dy}{dt}$.
- $6t$
- $\\frac{6t}{t^2+9}$
- $-6t$
- $\\frac{3t}{t^2+9}$
- Evaluate the definite integral $\\int_{1}^{4} (3x^2 - 2x + 1.25) dx$.
- $50.5$
- $51.5$
- $52.5$
- $53.5$
- Evaluate the indefinite integral $\\int x \\cos x dx$.
- $x \\sin x + \\cos x + C$
- $x \\sin x - \\cos x + C$
- $\\sin x + C$
- $x \\sin x \\cos x + C$
- Evaluate the indefinite integral $\\int \\ln x dx$.
- $x \\ln x + C$
- $x \\ln x + 1/x + C$
- $x(\\ln x - 1) + C$
- $\\ln x - x + C$
- Evaluate the integral $\\int \\frac{2}{2x+3} dx$.
- $\\ln|2x+3| + C$
- $2\\ln|2x+3| + C$
- $\\frac{1}{2}\\ln|2x+3| + C$
- $\\frac{-2}{(2x+3)^2} + C$
- If $y = \\frac{x+\\sin x}{x-\\cos x}$, find $\\frac{dy}{dx}$.
- $\\frac{x \\cos x - x \\sin x - \\cos x - \\sin x - 1}{(x - \\cos x)^2}$
- $\\frac{x \\sin x - x \\cos x - \\cos x + \\sin x + 1}{(x - \\cos x)^2}$
- $\\frac{x \\cos x + x \\sin x + \\cos x - \\sin x - 1}{(x - \\cos x)^2}$
- $\\frac{\\cos x - \\sin x - 1}{(x - \\cos x)^2}$
- Write the second term of the Maclaurin series expansion for $f(x) = e^x - 2$.
- $x$
- $-1$
- $\\frac{x^2}{2}$
- $1$
- If $y = 3^{3x}$, find $\\frac{dy}{dx}$.
- $3^{3x+1} \\ln 3$
- $3x \\cdot 3^{3x-1}$
- $3^{3x} \\ln 3$
- $3 \\ln 3$
- Evaluate the indefinite integral $\\int e^x(1+e^x)^2 dx$.
- $\\frac{(1+e^x)^3}{3} + C$
- $\\frac{(1+e^x)^2}{2} + C$
- $e^x(1+e^x)^3 + C$
- $\\frac{e^x(1+e^x)^3}{3} + C$
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