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Chapter 4 Lab Test Review
The following exercises are provided as examples of potential questions on the final lab test at the end of the semester.
Exercises Exercises
Riemann Sums 1. Assign the function
\(f(x) = x \lrp{1-x^2 { e}^{-\frac{1}{6}x^2}}\) in Maple.
(a) Evaluate the Riemann sum over the interval
\(\lrb{-4,2}\) using the
method=lower option with
\(8\) partitions.
(b) Evaluate the Riemann sum over the interval
\(\lrb{-4,2}\) using the
method=upper option with
\(8\) partitions.
2.
Assign the following function in Maple:
\begin{equation*}
g(x) = \sqrt{x} \sin(x)\text{.}
\end{equation*}
(a) Give an approximate value of
\(\dint_{1}^{8} g(x)~dx\) using the midpoint rule and
\(10\) partitions.
(b) Give an approximate value of
\(\dint_{1}^{8} g(x)~dx\) using the trapezoid rule and
\(10\) partitions.
(c) Give an approximate value of
\(\dint_{1}^{8} g(x)~dx\) using Simpson’s rule and
\(10\) partitions.
(d) Give the exact value of the definite integral
\(\dint_{1}^{8} g(x)~dx\) and evaluate as a decimal with 15 digits.
Integral Approximation Techniques 3.
Assign the function
\begin{equation*}
f(x) = \dint_{0}^{x} 10 { e}^{-0.5 t} \sin(t) \: dt
\end{equation*}
in Maple and plot it over the interval \([0,10]\text{.}\)
(a) What is the derivative,
\(f'(x)\text{?}\)
Solution .
\(10 {\mathrm e}^{- 0.5 x} \sin\! \left(x\right)\)
(b) At what value in the interval
\([0,10]\) does
\(f(x)\) reach its maximum?
(c) What is the maximum value of
\(f(x)\) over the interval
\([0,10]\text{?}\)
Areas Between Curves 4.
Assign the following function in Maple:
\begin{equation*}
h(x) = \dfrac{2x}{x^2+6} \text{.}
\end{equation*}
(a) Find the
net area bounded by
\(h(x)\) and the
\(x\) –axis on the interval
\(\lrb{-2,6}\text{.}\)
(b) Find the
total area bounded by
\(h(x)\) and the
\(x\) –axis on the interval
\(\lrb{-2,6}\text{.}\)
5. Plot the region between the curves
\(f(x) = \tan^2(x)\) and
\(g(x) = \sqrt{x}\) and compute the area to
\(15\) digits.
Average Value of a Function 6. Find the average value of
\(f(x) = 2\sin(x) - \sin(2x)\) on the interval
\([0,\pi]\text{.}\)
Volumes of Revolution 7.
Find the volume of the egg-shaped solid obtained by revolving the region bounded by the implicit curve
\begin{equation*}
4x^2 + y^2 = 12
\end{equation*}
about the \(x\) -axis.
8. Find the volume of the solid obtained by revolving the region bounded by the curves
\(y^2 - x^2 = 1\) and
\(y=2\) about the
\(x\) -axis.
Arc Length 9. Determine the arc length of the curve
\(f(x) = x\sqrt[3]{4-x}\) over the interval
\([0,4]\) to
\(15\) digits.
Infinite Integrals and Probability 10.
At an annual triathlon, the finishing times for male athletes can be modeled by the probability density function
\begin{equation*}
p(x) = \dfrac{1}{\sigma \sqrt{2\pi}} { e}^{-(x-\mu)^2/2\sigma^2} \text{,}
\end{equation*}
where \(\mu = 4313\) seconds (the average finish time) and \(\sigma = 583\) seconds (the standard deviation of finish times). Assign this function in Maple using the specified values of \(\mu\) and \(\sigma\) and plot the function over the interval \([0,6500]\text{.}\)
(a) What is the probability that a male athlete finishes the triathlon in under
\(3600\) seconds (
\(1\) hour)?
(b) What is the probability that a male athlete finishes the triathlon in over
\(4200\) seconds (
\(1\) hour
\(10\) min)?
(c) What is the probability that a male athlete takes between
\(3600\) and
\(5400\) seconds to finish the triathlon (
\(1\) hour to
\(1.5\) hours)?
Differential Equations 11.
After \(500\) fish are introduced to a lake, the rate of growth for the population of fish is given by the differential equation
\begin{equation*}
\frac{dN}{dt} = \dfrac{N\lrp{7000 - N}}{10000} \text{,}
\end{equation*}
where \(N=N(t)\) is the population of fish after \(t\) years. Define this differential equation in Maple.
(a) Use
dsolve() to give the solution to the differential equation for
\(N(t)\) using the initial condition
\(N(0)=500\text{.}\)
Solution .
\(N\! \left(t\right)=\frac{7000}{1+13 {\mathrm e}^{-7 t/10}}\)
(b) How many fish will there be after
\(6\) years (to the nearest fish)?
(c) What does the population of fish approach after a long time?
Hint .
You may need to take the limit as
\(t\) tends to infinity or determine the behaviour using a plot.
Solution .
Taylor Series 12. Find the Taylor series expansion of
\(f(x) = { e}^{2x} \tan(x)\) centred at
\(x=0\) (Maclaurin series) and give the coefficient of the
\(x^8\) term.
