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Active Calculus 2nd Ed

Matthew Boelkins

Section 9.1 Review of Prerequsites for Calculus I

Exercises Exercises

1.

For each of the following pairs of lines, find the slope, \(m\text{,}\) through the points or if undefined, type undefined.
  1. \({\left(-10,-4\right)}\) and \({\left(-10,-9\right)}\)
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    \(m=\)
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  2. \({\left(-4,-10\right)}\) and \({\left(0,-3\right)}\)
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    \(m=\)
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  3. \({\left(-8,10\right)}\) and \({\left(8,10\right)}\)
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    \(m=\)
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2.

The equation of the line that goes through the points \(( -4 ,7 )\) and \(( 10 ,-10 )\) can be written in the form \(y = mx+b\) where
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\(m =\) and \(b =\)
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3.

Find all real numbers \(x\) which satisfy the equation.
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\begin{equation*} 6 x^6 - 4 x^2 = 0 \end{equation*}
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Answer:
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Note: If there is more than one answer, write them separated by commas (e.g., 1, 2). Do not list individual values of \(x\) more than once.
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4.

If \(f(x)=x^{2} +7\text{,}\) find and simplify the following:
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(a) \(f(t+2) =\)
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(b) \(f(t^5+2) =\)
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(c) \(f(5) =\)
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(d) \(5 f(t) =\)
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(e) \((f(t))^2+2 =\)
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5.

Express the equation in exponential form
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(a) \(\log_{16} 2 = \frac{1}{4}\text{.}\)
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That is, write your answer in the form \(A^B=C\text{.}\) Then
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A=
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B=
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C=
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(b) \(\log_2\frac{1}{16} = -4\text{.}\)
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That is, write your answer in the form \(D^E=F\text{.}\) Then
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D=
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E=
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F=
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6.

The velocity (in ft/s) of a sky diver \(t\) seconds after jumping is given by
\begin{equation*} v(t) = 85 (1-e^{-0.1 t}) \end{equation*}
After how many seconds is the velocity 75 ft/s?
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seconds
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7.

Refer to the right triangle in the figure. Click on the picture to see it more clearly.
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If , \(BC=10\) and the angle \(\beta=70 ^\circ\text{,}\) find any missing angles or sides. Give your answer to at least 3 decimal digits.
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AB =
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AC =
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\(\alpha\)=
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8.

Click on the graph to view a larger graph
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For the given angle \(x\) in the triangle given in the graph
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\(\sin x=\) ;
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\(\cos x=\) ;
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\(\tan x=\) ;
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\(\cot x=\) ;
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\(\sec x=\) ;
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\(\csc x=\) ;
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9.

Solve the following equations in the interval [0,2\(\pi\)].
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Note: Give the answer as a multiple of \(\pi\text{.}\) Do not use decimal numbers. The answer should be a fraction or an integer. Note that \(\pi\) is already included in the answer so you just have to enter the appropriate multiple. E.g. if the answer is \(\pi/2\) you should enter 1/2. If there is more than one answer enter them separated by commas.
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\(\sin(t)= -\frac{\sqrt{2}}{2}\)
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\(t=\) \(\pi\)
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\(\sin(t)= -\frac{\sqrt{3}}{2}\)
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\(t=\) \(\pi\)
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10.

Solve the following equations in the interval [0, 2 \(\pi\)].
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Note: Give the answer as a multiple of \(\pi\text{.}\) Do not use decimal numbers. The answer should be a fraction or an integer. Note that \(\pi\) is already included in the answer so you just have to enter the appropriate multiple. E.g. if the answer is \(\pi/2\) you should enter 1/2. If there is more than one answer enter them separated by commas.
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\(\tan(t)=-{\sqrt{3}}\)
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\(t =\) \(\pi\)
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\(\tan(t)=-1\)
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\(t =\) \(\pi\)
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