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Section 1: Functions of Several Variables
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Engineering Mathematics I
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Practice Problems for Exam 1
Practice Problems for Exam 1
Directions.
The following are review problems. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. You can also follow the link for the full video page to find related videos.
Find the scalar and vector projections of \(\langle -3,1\rangle\) onto \(\langle 2,5\rangle\).
Answer
Scalar Projection: \(-\dfrac{1}{\sqrt{29}}\)
Vector Projection: \(\left\langle -\dfrac{2}{29}, -\dfrac{5}{29}\right\rangle\)
Video
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MLC WIR 20B M151 week3 #01
Vector
a
starts at the point \((-3,1)\) and ends at the point \((6,3)\). Find a unit vector in the direction of
a
.
Answer
\(\left\langle \dfrac{9}{\sqrt{85}}, \dfrac{2}{\sqrt{85}}\right\rangle\)
Video
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MLC WIR 20B M151 week3 #02
Determine a vector equation of the straight line which passes through the point \((1,-1)\) and is parallel to the vector \(\langle -2,3\rangle\).
Answer
\(\left\langle 1-2t, -1+3t \right\rangle\)
Video
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MLC WIR 20B M151 week3 #03
Consider the curve \(x(t)=t-2, y(t)=t^2-3\).
Is the point \((4,40)\) on the graph of the curve?
Eliminate the parameter to find a Cartesian equation.
Sketch the curve.
Answer
No
\(y=(x+2)^2-3\)
See Video
Video
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MLC WIR 20B M151 week3 #04
Find a Cartesian equation for the parametric curve given by the equation \(x=\dfrac{\cos t}{2}\), \(y=\sin t\), \(0\leq t \leq 2\pi\). Describe the direction of the motion as \(t\) increases.
Answer
\(4x^2+y^2=1\)
Video
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MLC WIR 20B M151 week3 #05
Find the angle \(\angle ABC\) of the triangle with the vertices, \(A(3,0)\), \(B(0,3)\), \(C(5,4)\).
Answer
\(\theta=\cos^{-1} \left( \dfrac{2}{\sqrt{13}}\right)\)
Video
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MLC WIR 20B M151 week3 #06
Find the distance from the point \((1,2)\) to the line \(y=3x-4\).
Answer
\(\dfrac{3}{\sqrt{10}}\)
Video
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MLC WIR 20B M151 week3 #07
Two forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\) with magnitudes 10 lb and 12 lb, respectively, act on an object at a point \(P\) as shown in the figure. Find the resultant force \(\mathbf{F}\) acting at \(P\) as well as its magnitude and its direction.
Answer
Magnitude: \(\sqrt{\left(-5\sqrt{2}+6\sqrt{3}\right)^2 + \left( 5\sqrt{2}+6\right)^2}\)
Direction: \(\theta = \tan^{-1}\left(\dfrac{5\sqrt{2}+6}{-5\sqrt{2}+6\sqrt{3}}\right)\)
Video
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MLC WIR 20B M151 week3 #08
A constant force with the vector representation \(\mathbf{F}=\langle 10, 18\rangle\) moves an object along a straight line form the point \((2,3)\) to the point \((4,9)\). Find the work done, if the distance is measure in meters and the magnitude of the force is measured in Newtons.
Answer
128 J
Video
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MLC WIR 20B M151 week3 #09
A woman exerts a horizontal force of 65 lb on a crate as she pushes it up a ramp that is 20 ft long and inclined at an angle of \(30^\circ\) above the horizontal. Find the work done on the box.
Answer
\(650\sqrt{3}\) ft\(\cdot\)lb
Video
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MLC WIR 20B M151 week3 #10
Find the vector equation of the line perpendicular to the vector \(\langle 3,5\rangle\) and passing through the point \((5,1)\).
Answer
\(\mathbf{r}(t)=\langle 5-5t, 1+3t\rangle\)
Video
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MLC WIR 20B M151 week3 #11
Consider the line \(x=8-2t\), \(y=14+7t\).
Find a vector perpendicular to the line
Find the Cartesian form.
Sketch the graph.
Answer
\(\langle-7,-2\rangle\) or \(\langle7,2\rangle\)
\(y=-\dfrac{7}{2}x+42\)
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Video
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MLC WIR 20B M151 week3 #12
Simplify.
\(\tan \left(\arccos\left( \dfrac{1}{4}\right)\right)\)
\(\sin(\arctan(2))\)
\(\tan(\arcsin(3x))\)
Answer
\(\sqrt{15}\)
\(\dfrac{2}{\sqrt{5}}\)
\(\dfrac{3x}{\sqrt{1-9x^2}}\)
Video
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MLC WIR 20B M151 week3 #13
Evaluate the limit.
\(\displaystyle \lim_{x\rightarrow 1} \dfrac{x^2-x-2}{x+1}\)
\(\displaystyle \lim_{t\rightarrow 0}\dfrac{\sqrt{2-t}-\sqrt{2}}{t}\)
\(\displaystyle \lim_{h\rightarrow 0}\dfrac{(3+h)^{-1}-3^{-1}}{h}\)
\(\displaystyle \lim_{x\rightarrow -4^{-}} \dfrac{|x+4|}{x+4}\)
\(\displaystyle \lim_{x\rightarrow -1^{-}} \dfrac{x-2}{x+1}\)
\(\displaystyle \lim_{x\rightarrow -\infty} \dfrac{-x^3+2x^2-4x}{8+4x^2-5x^3}\)
\(\displaystyle \lim_{x\rightarrow \infty} \dfrac{e^x+2e^{-x}}{2e^x-e^{-x}}\)
\(\displaystyle \lim_{x\rightarrow \infty}\ln \left( \dfrac{x}{x^2-3x}\right)\)
\(\displaystyle \lim_{x\rightarrow \infty} \arctan\left(e^x\right)\)
\(\displaystyle \lim_{x\rightarrow \infty} \arctan(\ln x)\)
\(\displaystyle\lim_{x\rightarrow 0^+} \arctan (\ln x)\)
Answer
\(-1\)
\(-\dfrac{1}{2\sqrt{2}}\)
\(-\dfrac{1}{9}\)
\(-1\)
\(\infty\)
\(\dfrac{1}{5}\)
\(\dfrac{1}{2}\)
\(-\infty\)
\(\dfrac{\pi}{2}\)
\(\dfrac{\pi}{2}\)
\(-\dfrac{\pi}{2}\)
Video
Given \(-2x-2\leq f(x) \leq \dfrac{1}{2} x^2\), compute \(\displaystyle \lim_{x\rightarrow -2} f(x)\).
Answer
2
Video
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MLC WIR 20B M151 week3 #15
Find the vertical and horizontal asymptotes of \(f(x)=\dfrac{(x-1)(x+3)}{x^2-1}\).
Answer
Vertical Asymptote: \(x=-1\)
Horizontal Asymptote: \(y=1\)
Video
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MLC WIR 20B M151 week3 #16
Given the graph of \(f(x)\) below, sketch the graph of the derivative.
Answer
See the Video
Video
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MLC WIR 20B M151 week3 #17
Given the function below, find the value of \(a\) which makes the function continuous everywhere.\[f(x)=\left\{ \begin{array}{cc} x-4a& \textrm{if } x<-2\\ax^2 & \textrm{if } x\geq -2\end{array} \right.\]
Answer
\(a=-\dfrac{1}{4}\)
Video
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MLC WIR 20B M151 week3 #18
Which of the following intervals must contian a solution to the equation \(2x^3+16x+3=22\)?
\([-2,-1]\)
\([-1,0]\)
\([0,1]\)
\([1,2]\)
\([2,3]\)
Answer
d. \([1,2]\)
Video
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MLC WIR 20B M151 week3 #19
Find the average rate of change of \(f(x)=x^2+6\) from \(x=-3\) to \(x=1\).
Answer
-2
Video
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MLC WIR 20B M151 week3 #20
Find \(f'(x)\) using the limit definition of the derivative for \(f(x)=3x^2-4\).
Answer
\(f'(x)=6x\)
Video
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MLC WIR 20B M151 week3 #21
Find \(f'(x)\) using the limit definition of the derivative for \(f(x)=\sqrt{3x+1}\).
Answer
\(f'(x)=\dfrac{3}{2\sqrt{3x+1}}\)
Video
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MLC WIR 20B M151 week3 #22
Find \(f'(x)\) using the limit definition of the derivative for \(f(x)=\dfrac{-2}{x+2}\).
Answer
\(f'(x)=\dfrac{2}{(x+2)^2}\)
Video
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MLC WIR 20B M151 week3 #23