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Chapter 2 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
The Basics 1. Evaluate
\(\sqrt[3]{\dfrac{5.0^2 - 3.0}{1.5 + 9.0^2}}\) to
\(15\) digits.
2. Give the numerical value of
\({ e}^2\) to
\(15\) digits.
3.
Factor the given cubic to find its roots.
\begin{equation*}
2x^3 - 7x^2 + 7x -2
\end{equation*}
Solution .
\(\left(x-1\right) \left(x-2\right) \left(2 x-1\right)\)
4. Find the point of intersection of the curves
\(y = { e}^x\) and
\(y = \frac{1}{x}\text{.}\) Give both the
\(x\) - and
\(y\) -coordinates to
\(15\) digits.
Solution .
\(x= 0.567143290409784, y= 1.76322283435190\)
Limits 5.
Find the left- and right-hand limits of
\begin{equation*}
\dfrac{x^2 |x-2|}{x-2}
\end{equation*}
at \(x=2\text{.}\) Does the two-sided limit at \(x=2\) exist? Explain.
Solution .
\(4\) from the right,
\(-4\) from the left. No, the two-sided limit at
\(x = 2\) does not exist because the left and right limits are different.
6.
Consider the function
\begin{equation*}
f(x)=\frac{\sin(x)}{x^2+1}+\arctan(x)\text{.}
\end{equation*}
Evaluate the following limits to \(15\) digits.
(a)
\(\dlim{x}{\infty} f(x)\)
(b)
\(\dlim{x}{-\infty} f(x)\)
Vertical and Horizontal Asymptotes 7. Sketch the plot of
\(y=\dfrac{ \sqrt{4x^2 + 1} }{2x - 1}\) and state the equations of the vertical and horizontal asymptotes.
Solution .
Vertical asymptote:
\(x=\frac{1}{2}\text{;}\) Horizontal asymptotes:
\(y=\pm 1\)
8. Find the horizontal asymptote of
\(\dfrac{5x^3 - 2x + 1}{1 - 8x^3}\text{.}\)
The Derivative of a Function 9.
Consider the function
\begin{equation*}
f(x) = (1-x^2) { e}^{-x^2/2}\text{.}
\end{equation*}
Find all critical numbers of \(f\) to \(15\) digits.
Solution .
\(0, 1.73205080756888,- 1.73205080756888\)
10.
Consider the function
\begin{equation*}
g(x) = \frac{x+2}{3\sqrt{x^2+5}}\text{.}
\end{equation*}
(a) Plot
\(g(x)\text{,}\) \(g'(x)\text{,}\) and
\(g''(x)\) on the same axes.
(b) Find the maximum value of
\(g(x)\text{.}\)
Solution .
Maximum value of
\(0.447213595499958\) at
\(x=2.5\text{.}\)
(c) Find the maximum value of
\(g'(x)\text{.}\)
Solution .
Maximum value of
\(0.166566474658809\) at
\(x=-0.57767710879358\text{.}\)
11.
The height in metres of a ball thrown from the top of a building is given by the function
\begin{equation*}
h(t) = -9.80t^2 + 5.00t + 40\text{.}
\end{equation*}
(a) Find the velocity of the ball at
\(t=2\) seconds.
(b) At what time does the ball reach its maximum height?
Solution .
\(t=0.255102040816327\) seconds
(c) What is the maximum height of the ball?
Solution .
\(40.6377551020408\) metres
Tangent Lines 12.
Given the function
\begin{equation*}
f(x) = { e}^x \cos x\text{,}
\end{equation*}
find the equation of the tangent line to \(f(x)\) at \(x=1\) in the form \(y = mx + b\text{.}\)
Solution .
\(y=-0.818661347262959 x + 2.28735528717885\)
13.
Given the function
\begin{equation*}
g(x) = x^{\ln x}\text{,}
\end{equation*}
find the equation of the tangent line to \(g(x)\) at \(x=5\) in the form \(y = mx + b\text{.}\)
Solution .
\(y=8.58386825467852 x - 29.5856982226701\)
Implicit Functions 14.
Given the Folium of Descartes,
\begin{equation*}
x^3 + y^3 = 6xy\text{,}
\end{equation*}
find the slopes of the tangent lines to the curve at \(x=2\text{.}\)
Solution .
\(-1.13715804260326, 0.742227198968559, 0.394930843634699\)
15.
Find all points on the curve
\begin{equation*}
(x^2+y^2)^2 = 16(y^2-x^2)
\end{equation*}
where the slope of the tangent line to the curve is equal to \(4\text{.}\)
Hint .
You may need to use the
fsolve() command and specify different intervals to find the required points.
Solution .
\((-1.35130857040332, -1.95308486464414)\text{,}\) \((-1.36235641252506, 2.87932257596956)\text{,}\) \((1.36235641252506, -2.87932257596956)\text{,}\) \((1.35130857040332, 1.95308486464414)\)
