# Diagnostic Quiz 2: Some calculus

Question

## Question 1

Find the value of the derivative $\phantom{\rule{2.77695pt}{0ex}}{f}^{\prime }\left(1\right)\phantom{\rule{2.77695pt}{0ex}}$ for the following function: $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}f\left(x\right)={x}^{2}+6x-5$
 a) $8$ b) $3$ c) $2$ d) $\frac{10}{3}$ e) $-3$

Your answer is correct.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.

## Question 2

Find the derivative of the following function: $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}f\left(x\right)=cosx-sinx$
 a) ${f}^{\prime }\left(x\right)=sinx-cosx\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$ b) ${f}^{\prime }\left(x\right)=sinx+cosx\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$ c) ${f}^{\prime }\left(x\right)=cosx-sinx\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$ d) ${f}^{\prime }\left(x\right)=-cosx-sinx\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$ e) ${f}^{\prime }\left(x\right)=-1-tanx\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Not correct. Choice (e) is false.

## Question 3

Find the derivative of the following function: $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}f\left(x\right)=\sqrt{x}$
 a) ${f}^{\prime }\left(x\right)=\frac{2}{\sqrt{x}}$ b) ${f}^{\prime }\left(x\right)=-\frac{2}{\sqrt{x}}$ c) ${f}^{\prime }\left(x\right)=\frac{1}{2\sqrt{x}}$ d) ${f}^{\prime }\left(x\right)=-\frac{1}{2\sqrt{x}}$ e) ${f}^{\prime }\left(x\right)=\frac{2}{3}{x}^{3∕2}$

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.

## Question 4

Find the derivative of the following function: $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}f\left(x\right)={e}^{3x}$
 a) ${f}^{\prime }\left(x\right)={e}^{3x}$ b) ${f}^{\prime }\left(x\right)=3x{e}^{3x}$ c) ${f}^{\prime }\left(x\right)=\frac{{e}^{3x}}{3}$ d) ${f}^{\prime }\left(x\right)={e}^{3x}+3$ e) ${f}^{\prime }\left(x\right)=3{e}^{3x}$

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Your answer is correct.

## Question 5

Find the derivative of the following function: $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}f\left(x\right)={x}^{2}lnx$
 a) ${f}^{\prime }\left(x\right)=x$ b) ${f}^{\prime }\left(x\right)=2x+\frac{1}{x}$ c) ${f}^{\prime }\left(x\right)=2x-\frac{1}{x}$ d) ${f}^{\prime }\left(x\right)=x\left(2lnx+1\right)$ e) ${f}^{\prime }\left(x\right)=2xlnx+\frac{1}{x}$

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Not correct. Choice (e) is false.

## Question 6

Find the derivative of the following function: $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}f\left(x\right)=\frac{x-1}{{x}^{2}+1}$
 a) ${f}^{\prime }\left(x\right)=\frac{1+2x-{x}^{2}}{{\left({x}^{2}+1\right)}^{2}}$ b) ${f}^{\prime }\left(x\right)=\frac{1+2x-{x}^{2}}{{x}^{2}+1}$ c) ${f}^{\prime }\left(x\right)=\frac{{\left(x+1\right)}^{2}}{{x}^{2}+1}$ d) ${f}^{\prime }\left(x\right)=\frac{{\left(x-1\right)}^{2}}{{x}^{2}+1}$ e) ${f}^{\prime }\left(x\right)=\frac{1-2x-{x}^{2}}{{\left({x}^{2}+1\right)}^{2}}$

Your answer is correct.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.

## Question 7

Find the equation of the tangent line to the curve $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}y={x}^{3}-x+4\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$ at the point $\phantom{\rule{2.77695pt}{0ex}}\left(-1,4\right)\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$.
 a) $y=-4x$ b) $y=-x+3$ c) $y=x+5$ d) $y=2x+6$ e) $y=-2x+2$

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Not correct. Choice (e) is false.

## Question 8

Find the maximum value of $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}f\left(x\right)={x}^{3}-3{x}^{2}-9x+2\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$ for $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}-2\le x\le 1\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$.
 a) $7$ b) $2$ c) $-9$ d) $0$ e) $\frac{15}{2}$

Your answer is correct.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.

## Question 9

Evaluate the definite integral $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\int }_{1}^{2}{x}^{2}+5x-1\phantom{\rule{0.3em}{0ex}}dx\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$.
 a) $\frac{23}{6}$ b) $\frac{53}{6}$ c) $\frac{17}{3}$ d) $9$ e) $\frac{55}{6}$

Not correct. Choice (a) is false.
Your answer is correct.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.

## Question 10

Find the indefinite integral $\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\int sin2x\phantom{\rule{0.3em}{0ex}}dx\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$.
 a) $cos2x+C$ b) $-2cos2x+C$ c) $\frac{cos2x}{2}+C$ d) $2cos2x+C$ e) $-\frac{cos2x}{2}+C$

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Your answer is correct.
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