This quiz tests the work covered in Lecture 13 and corresponds to Section 3.2 of the
textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et
al.).

There are more web quizzes at Wiley, select Section 2.

Section 3.6 of The Mathematics Learning Hub’s booklet on differentiation
Introduction to Differential Calculus covers differentiating the exponential
function.

Which of the following is the derivative of
$g\left(t\right)={5}^{t}\phantom{\rule{0.3em}{0ex}}?$ Exactly one option
must be correct)

Which of the following is the derivative of
$y=3{x}^{2}+2{e}^{x}\phantom{\rule{0.3em}{0ex}}?$ Exactly one option
must be correct)

Choice (a) is incorrect

Try again, you have not differentiated the first term.

Choice (b) is incorrect

Try again, you have not differentiated the first term correctly.

Choice (c) is incorrect

Try again, you have not differentiated the last term correctly.

Choice (d) is correct!

The
derivative of ${x}^{2}$
is $2x$ and the
derivative of ${e}^{x}$
is ${e}^{x}$
so $\frac{dy}{dx}=3\left(2x\right)+2\left({e}^{x}\right)=6x+2{e}^{x}\phantom{\rule{0.3em}{0ex}}.$

Which of the following is the equation of tangent to
$y=2-2{e}^{x}$ at
$x=1\phantom{\rule{0.3em}{0ex}}.$ Exactly one option must be
correct)

Choice (a) is incorrect

Try again, this is
equation of the tangent at $x=0\phantom{\rule{0.3em}{0ex}}.$

Choice (b) is incorrect

Try again, you have the correct gradient but the intercept is incorrect.

Choice (c) is incorrect

Try again, you seem to have confused the gradient and the intercept.

Choice (d) is correct!

$\frac{dy}{dx}=-2{e}^{x}$ so
${\left.\frac{dy}{dx}\right|}_{x=1}=-2e\phantom{\rule{0.3em}{0ex}}.$ The gradient of the tangent is therefore
$-2e$ and
$y=\left(-2e\right)x+b\phantom{\rule{0.3em}{0ex}}.$ At $x=1\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}y=2-2e$ and the
tangent must pass through this point so we substitute these values into the equation of the
line to find $b\phantom{\rule{0.3em}{0ex}}.$ $2-2e=-2e\left(1\right)+b\Rightarrow b=2$ and
$y=\left(-2e\right)x+2=2-\left(2e\right)x\phantom{\rule{0.3em}{0ex}}.$

We saw that if the population of a city increases at a rate which
is proportional to the current population and was 2 million in
1980 and 2.5 million in 1990 then the formula for the population
$t$ years after 1980
was given by $P\left(t\right)=2{e}^{0.0223t}\phantom{\rule{0.3em}{0ex}}.$
Which of the following correctly gives the population in the form
$P\left(t\right)={P}_{0}^{}{a}^{t}\phantom{\rule{0.3em}{0ex}},$ and the
growth rate of $t$
years after 1980, to 4 decimal places?
Exactly one option must be correct)

Choice (a) is incorrect

Try again, you do not have the correct growth rate.