Elementary Mathematics II — sodiomatics Past Questions
University of Ilorin MTH102: 32 questions.
Topics: Derivatives as limits of rate of change, differentiation techniques, maxima and minima., Functions of a real variable, graphs, limits, and continuity., Curve sketching, integration, definite integrals, reduction formulae.
- If $g(x) = 5x^2$, $g(x)$ is said to be what?
- Even function
- Symmetrical function
- Odd function
- None of the above
- Find the derivative of $y = \sin(x^3)$.
- $3x^2 \cos(x^3)$
- $3x^2 \sin(x^3)$
- $\cos(x^3)$
- $-\cos(x^3)$
- Find the derivative of $y = e^{-x^2} \sin(x^2)$.
- $2x e^{-x^2}(\cos(x^2) - \sin(x^2))$
- $-2x e^{-x^2}(\cos(x^2) + \sin(x^2))$
- $2x e^{-x^2}(\sin(x^2) - \cos(x^2))$
- $e^{-x^2}(\cos(x^2) - \sin(x^2))$
- Under what condition is a function said to be stationary?
- $\frac{dy}{dx} = 0$
- $\frac{dy}{dx} < -1$
- $\frac{dy}{dx} > 1$
- $\frac{dy}{dx} = 1$
- Find the derivative of $y = \frac{1}{\sqrt{2x+3}}$.
- $\frac{-1}{(2x+3)^{3/2}}$
- $\frac{1}{(2x+3)^{3/2}}$
- $\frac{-2}{(2x+3)^{3/2}}$
- $\frac{-1}{2(2x+3)^{3/2}}$
- Given $F(x) = 4x+5$, with $\alpha = 2$ and $\epsilon = \frac{1}{2}$, find the limit $L$ as $x \to \alpha$ and the corresponding maximum $\delta$.
- $L=13$ and $\delta=1/8$
- $L=13$ and $\delta=1/4$
- $L=8$ and $\delta=1/8$
- $L=13$ and $\delta=1/2$
- Given $F(x) = 5x+4$, where $\alpha = 6$ and $\epsilon = \frac{1}{10}$, find the maximum value of $\delta$.
- $1/50$
- $1/10$
- $1/20$
- $1/5$
- Find the differential (derivative) of $y = 4^{(x^2+1)^3}$.
- $4^{(x^2+1)^3} \cdot 6x(x^2+1)^2 \ln 4$
- $4^{(x^2+1)^3} \cdot 3(x^2+1)^2 \ln 4$
- $4^{(x^2+1)^3} \cdot 6x(x^2+1)^2$
- $4^{(x^2+1)^3} \cdot (x^2+1)^2 \ln 4$
- Find the domain of the function $f(x) = \sqrt{3-x}$.
- $x \le 3$
- $x < 3$
- $x \ge 3$
- $x > 3$
- Find the Maclaurin series expansion of $f(x) = \frac{1}{1-x}$.
- $1 + x + x^2 + x^3 + \dots$
- $1 - x + x^2 - x^3 + \dots$
- $1 + 2x + 3x^2 + 4x^3 + \dots$
- $x + x^2 + x^3 + x^4 + \dots$
- What is the 4th non-zero term of the Maclaurin series for $f(x) = \frac{1}{1-x^2}$?
- $x^6$
- $x^4$
- $x^3$
- $x^8$
- Given a piecewise function $f(x) = 5x+4$ for $x < 2$ and $f(x) = 3x-2$ for $x \ge 2$. What can be said about the limit as $x \to 2$?
- The limit does not exist
- The limit exists and equals 14
- The limit exists and equals 4
- The function is continuous at $x=2$
- Given $F(x) = \frac{x^2-1}{x-1}$, $\alpha = 3$, and $\epsilon = 0.01$. Find the maximum value of $\delta$.
- $0.01$
- $0.02$
- $0.1$
- $0.001$
- If $f(x) = \frac{x+2}{x-3}$, find its inverse $f^{-1}(x)$.
- $\frac{3x+2}{x-1}$
- $\frac{2x+3}{x-1}$
- $\frac{3x-2}{x+1}$
- $\frac{x-3}{x+2}$
- If $F(x) = |x|$, which of the following is true?
- It is continuous everywhere
- It is not continuous at $x=0$
- It is differentiable everywhere
- It is discontinuous everywhere
- If $f(x) = x^3$, then the function is:
- Odd, and the graph is symmetrical about the origin
- Odd, and the graph is symmetrical about the y-axis
- Even, and the graph is symmetrical about the y-axis
- Even, and the graph is symmetrical about the origin
- Find $\frac{dy}{dx}$ of $y = \tan^{-1}(5x^2)$.
- $\frac{10x}{1+25x^4}$
- $\frac{10x}{1+5x^4}$
- $\frac{5x}{1+25x^4}$
- $\frac{10x}{1-25x^4}$
- Find $\frac{dy}{dx}$ if $x^2 + y^2 = \cos x$.
- $\frac{-\sin x - 2x}{2y}$
- $\frac{\cos x - 2x}{2y}$
- $\frac{\sin x + 2x}{2y}$
- $\frac{-\cos x - 2x}{2y}$
- Find the stationary points of $y = x + \frac{9}{x}$.
- $\pm 3$
- $\pm 1/3$
- $\pm 9$
- $\pm \sqrt{3}$
- If $y = \frac{m}{m^2+1}$ and $x = m^2+1$, find $\frac{dy}{dx}$.
- $\frac{1-m^2}{2m(m^2+1)^2}$
- $\frac{1+m^2}{2m(m^2+1)^2}$
- $\frac{1-m^2}{m(m^2+1)^2}$
- $\frac{1-m^2}{2m}$
- Find $\frac{dy}{dx}$ of $y = \tan x + \frac{\sqrt{x+1}}{\sqrt{x}}$.
- $\sec^2 x - \frac{1}{2\sqrt{x^3(x+1)}}$
- $\sec^2 x + \frac{1}{2\sqrt{x(x+1)}}$
- $\sec^2 x - \frac{1}{2\sqrt{x(x+1)}}$
- $\csc^2 x - \frac{1}{2\sqrt{x^3(x+1)}}$
- Find the stationary points of $y = 3x^4 - 8x^3 + 6x^2 + 10$.
- $0 \text{ and } 1$
- $1 \text{ and } -1$
- $0 \text{ and } 2$
- $1 \text{ and } 2$
- Find the derivative of $y = \sin^{-1} x$.
- $\frac{1}{\sqrt{1-x^2}}$
- $\frac{-1}{\sqrt{1-x^2}}$
- $\frac{1}{1+x^2}$
- $\frac{1}{\sqrt{x^2-1}}$
- Which of the following conditions represents a one-to-one (injective) function?
- If $f(a_1) = f(a_2)$ then $a_1 = a_2$
- If $f(a_1) = f(a_2)$ then $a_1 \neq a_2$
- If $f(a_1) \neq f(a_2)$ then $a_1 = a_2$
- None of the above
- Find the points of inflexion for the curve $y = 3x^4 - 8x^3 + 6x^2 + 5$.
- $x = 1/3 \text{ and } x = 1$
- $x = -1/3 \text{ and } x = 1$
- $x = 1/3 \text{ and } x = -1$
- $x = 3 \text{ and } x = 1$
- Find the stationary points of $y = x^3 - 2x^2 + x + 3$.
- $x = 1 \text{ and } x = 1/3$
- $x = -1 \text{ and } x = 1/3$
- $x = 1 \text{ and } x = -1/3$
- $x = 3 \text{ and } x = 1$
- Determine the nature of the turning point on the curve $y = x(1-x^2)$ at $x = \frac{-1}{\sqrt{3}}$.
- Minimum
- Maximum
- Point of Inflexion
- Undefined
- Expand the function $f(x) = e^x$ using Maclaurin series to the 4th term.
- $1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}$
- $1 - x + \frac{x^2}{2!} - \frac{x^3}{3!}$
- $x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!}$
- $1 + 2x + 3x^2 + 4x^3$
- Find $\frac{dy}{dx}$ if $y = a^{\sin 2x}$.
- $2a^{\sin 2x} \cos 2x \ln a$
- $a^{\sin 2x} \cos 2x \ln a$
- $2a^{\sin 2x} \sin 2x \ln a$
- $2a^{\sin 2x} \cos 2x$
- Find the x-values of the turning points when $y = 4x^3 - 3x^2 - 6x + 1$.
- $1 \text{ and } -1/2$
- $-1 \text{ and } 1/2$
- $1 \text{ and } 2$
- $-1/2 \text{ and } 2$
- Find $\frac{dy}{dx}$ if $y = \ln(x^2 + 3x + 5)$.
- $\frac{2x+3}{x^2+3x+5}$
- $\frac{1}{x^2+3x+5}$
- $\frac{2x}{x^2+3x+5}$
- $\frac{2x+3}{(x^2+3x+5)^2}$
- Evaluate $\lim_{x \to 2} \frac{x-2}{x^2-4}$.
- $1/4$
- $1/2$
- $0$
- $\infty$
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