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Nelson Calculus and Vectors 12

Chris Kirkpatrick, Peter Crippin

Chapter 5

Rate-Of-Change Models in Microbiology


Problem 1

Why can you not use the power rule for derivatives to differentiate $y=e^{x} ?$

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Problem 2

Differentiate each of the following:
a. $y=e^{3 x}$
b. $s=e^{3 t-5}$
c. $y=2 e^{10 t}$
d. $y=e^{-3 x}$
e. $y=e^{5-6 x+x^{2}}$
f. $y=e^{\sqrt{x}}$

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Problem 3

Determine the derivative of each of the following:
a. $y=2 e^{x^{3}}$
b. $y=x e^{3 x}$
c. $f(x)=\frac{e^{-x^{3}}}{x}$
d. $f(x)=\sqrt{x} e^{x}$
e. $h(t)=e t^{2}+3 e^{-t}$
f. $g(t)=\frac{e^{2 t}}{1+e^{2 t}}$

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Problem 4

a. If $f(x)=\frac{1}{3}\left(e^{3 x}+e^{-3 x}\right),$ calculate $f^{\prime}(1)$.
b. If $f(x)=e^{-\left(\frac{1}{a+1}\right)},$ calculate $f^{\prime}(0)$.
c. If $h(z)=z^{2}\left(1+e^{-z}\right),$ calculate $h^{\prime}(-1)$.

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Problem 5

a. Determine the equation of the tangent to the curve defined by $y=\frac{2 e^{x}}{1+e^{x}}$
at the point (0,1).
b. Use graphing technology to graph the function in part a., and draw the tangent at (0,1).
c. Compare the equation in part a. with the equation generated by graphing technology. Do they agree?

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Problem 6

Determine the equation of the tangent to the curve $y=e^{-x}$ at the point where $x=-1 .$ Graph the original curve and the tangent.

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Problem 7

Determine the equation of the tangent to the curve defined by $y=x e^{-x}$ at the point $A\left(1, e^{-1}\right)$.

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Problem 8

Determine the coordinates of all points at which the tangent to the curve defined by $y=x^{2} e^{-x}$ is horizontal.

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Problem 9

If $y=\frac{5}{2}\left(e^{\frac{x}{5}}+e^{-\frac{x}{5}}\right),$ prove that $y^{\prime \prime}=\frac{y}{25}$.

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Problem 10

a. For the function $y=e^{-3 x},$ determine $\frac{d y}{d x}, \frac{d^{2} y}{d x^{2}},$ and $\frac{d^{3} y}{d x^{3}}$.
b. From the pattern in part a., state the value of $\frac{d^{n} y}{d x^{n}}$.

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Problem 11

Determine the first and second derivatives of each function.
$\begin{array}{lll}\text { a. } y=-3 e^{x} & \text { b. } y=x e^{2 x} & \text { c. } y=e^{x}(4-x)\end{array}$

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Problem 12

The number, $N$, of bacteria in a culture at time $t,$ in hours, is $N(t)=1000\left[30+e^{-\frac{1}{5}}\right]$
a. What is the initial number of bacteria in the culture?
b. Determine the rate of change in the number of bacteria at time $t$.
c. How fast is the number of bacteria changing when $t=20 ?$
d. Determine the largest number of bacteria in the culture during the interval $0 \leq t \leq 50$.
e. What is happening to the number of bacteria in the culture as time passes?

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Problem 13

The distance $s,$ in metres, fallen by a skydiver $t$ seconds after jumping (and before the parachute opens) is $s=160\left(\frac{1}{4} t-1+e^{-\frac{t}{4}}\right)$.
a. Determine the velocity, $v,$ at time $t$.
b. Show that acceleration is given by $a=10-\frac{1}{4} v$.
c. Determine $v_{r}=\lim _{t \rightarrow \infty} v .$ This is the "terminal" velocity, the constant velocity attained when the air resistance balances the force of gravity.
d. At what time is the velocity $95 \%$ of the terminal velocity? How far has the skydiver fallen at that time?

Bobby B.
University of North Texas

Problem 14

a. Use a table of values and successive approximation to evaluate each of the following:
i. $\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)$
ii. $\lim _{x \rightarrow 0}(1+x)^{\frac{1}{x}}$
b. Discuss your results.

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Problem 15

Use the definition of the derivative to evaluate each limit.
a. $\lim _{h \rightarrow 0} \frac{e^{h}-1}{h}$
b. $\lim _{h \rightarrow 0} \frac{e^{2+h}-e^{2}}{h}$

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Problem 16

For what values of $m$ does the function $y=A e^{m t}$ satisfy the following equation? $\frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}-6 y=0$

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Problem 17

The hyperbolic functions are defined as $\sinh x=\frac{1}{2}\left(e^{x}-e^{-x}\right)$ and $\cosh x=\frac{1}{2}\left(e^{x}+e^{-x}\right)$.
a. Prove $\frac{d(\sinh x)}{d x}=\cosh x$.
b. Prove $\frac{d(\cosh x)}{d x}=\sinh x$.
c. Prove $\frac{d(\tanh x)}{d x}=\frac{1}{(\cosh x)^{2}}$ if $\tanh x=\frac{\sinh x}{\cosh x}$.

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