Difference between revisions of "009A Sample Midterm 1"

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<span class="exam"> Find the following limits:
 
<span class="exam"> Find the following limits:
  
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<span class="exam">(a) Find <math>\lim _{x\rightarrow 2} g(x),</math> provided that <math>\lim _{x\rightarrow 2} \bigg[\frac{4-g(x)}{x}\bigg]=5</math>
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<span class="exam">(a) Find &nbsp;<math style="vertical-align: -13px">\lim _{x\rightarrow 2} g(x),</math>&nbsp; provided that &nbsp;<math style="vertical-align: -15px">\lim _{x\rightarrow 2} \bigg[\frac{4-g(x)}{x}\bigg]=5.</math>
  
−
<span class="exam">(b) Find <math>\lim _{x\rightarrow 0} \frac{\sin(4x)}{5x} </math>
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<span class="exam">(b) Find &nbsp;<math style="vertical-align: -14px">\lim _{x\rightarrow 0} \frac{\sin(4x)}{5x} </math>
  
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<span class="exam">(c) Evaluate <math>\lim _{x\rightarrow -3^+} \frac{x}{x^2-9} </math>
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<span class="exam">(c) Evaluate &nbsp;<math style="vertical-align: -14px">\lim _{x\rightarrow -3^+} \frac{x}{x^2-9} </math>
  
 
== [[009A_Sample Midterm 1,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
 
== [[009A_Sample Midterm 1,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
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<span class="exam">Consider the following function <math> f:</math>
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<span class="exam">Suppose the size of a population at time &nbsp;<math style="vertical-align: 0px">t</math>&nbsp; is given by
 +
 
 +
::<math>N(t)=\frac{1000t}{5+t},~t\ge 0.</math>
 +
 
 +
<span class="exam">(a) Determine the size of the population as &nbsp;<math style="vertical-align: -1px">t\rightarrow \infty.</math>&nbsp; We call this the limiting population size.
 +
 
 +
<span class="exam">(b) Show that at time &nbsp;<math style="vertical-align: -4px">t=5,</math>&nbsp; the size of the population is half its limiting size.
 +
 
 +
== [[009A_Sample Midterm 1,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
 +
<span class="exam">Consider the following function &nbsp;<math style="vertical-align: -5px"> f:</math>
 
::<math>f(x) = \left\{
 
::<math>f(x) = \left\{
 
     \begin{array}{lr}
 
     \begin{array}{lr}
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</math>
 
</math>
  
−
<span class="exam">(a) Find <math> \lim_{x\rightarrow 1^-} f(x).</math>
+
<span class="exam">(a) Find &nbsp;<math style="vertical-align: -15px"> \lim_{x\rightarrow 1^-} f(x).</math>
  
−
<span class="exam">(b) Find <math> \lim_{x\rightarrow 1^+} f(x).</math>
+
<span class="exam">(b) Find &nbsp;<math style="vertical-align: -15px"> \lim_{x\rightarrow 1^+} f(x).</math>
  
−
<span class="exam">(c) Find <math> \lim_{x\rightarrow 1} f(x).</math>
+
<span class="exam">(c) Find &nbsp;<math style="vertical-align: -13px"> \lim_{x\rightarrow 1} f(x).</math>
  
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<span class="exam">(d) Is <math>f</math> continuous at <math>x=1?</math> Briefly explain.
+
<span class="exam">(d) Is &nbsp;<math style="vertical-align: -5px">f</math>&nbsp; continuous at &nbsp;<math style="vertical-align: -1px">x=1?</math>&nbsp; Briefly explain.
  
−
== [[009A_Sample Midterm 1,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
+
== [[009A_Sample Midterm 1,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
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<span class="exam"> Let <math>y=\sqrt{3x-5}.</math>
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<span class="exam"> Let &nbsp;<math style="vertical-align: -5px">y=\sqrt{3x-5}.</math>
  
−
::<span class="exam">a) Use the definition of the derivative to compute <math>\frac{dy}{dx}</math> for <math>y=\sqrt{3x-5}.</math>
+
<span class="exam">(a) Use the definition of the derivative to compute &nbsp; <math>\frac{dy}{dx}</math> &nbsp; for &nbsp;<math style="vertical-align: -5px">y=\sqrt{3x-5}.</math>
−
::<span class="exam">b) Find the equation of the tangent line to <math>y=\sqrt{3x-5}</math> at <math>(2,1).</math>
 
  
−
== [[009A_Sample Midterm 1,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
+
<span class="exam">(b) Find the equation of the tangent line to &nbsp;<math style="vertical-align: -5px">y=\sqrt{3x-5}</math>&nbsp; at &nbsp;<math style="vertical-align: -5px">(2,1).</math>
−
<span class="exam"> Find the derivatives of the following functions. Do not simplify.
 
−
 
 
−
::<span class="exam">a) <math>f(x)=\sqrt{x}(x^2+2)</math>
 
−
::<span class="exam">b) <math>g(x)=\frac{x+3}{x^{\frac{3}{2}}+2}</math> where <math>x>0</math>
 
−
::<span class="exam">c) <math>h(x)=\frac{e^{-5x^3}}{\sqrt{x^2+1}}</math>
 
  
 
== [[009A_Sample Midterm 1,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
 
== [[009A_Sample Midterm 1,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
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<span class="exam"> The displacement from equilibrium of an object in harmonic motion on the end of a spring is:
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<span class="exam"> Find the derivatives of the following functions. Do not simplify.
  
−
::::::<span class="exam"><math>y=\frac{1}{3}\cos(12t)-\frac{1}{4}\sin(12t)</math>
+
<span class="exam">(a) &nbsp; <math style="vertical-align: -5px">f(x)=\sqrt{x}(x^2+2)</math>
  
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<span class="exam">where <math>y</math> is measured in feet and <math>t</math> is the time in seconds.
+
<span class="exam">(b) &nbsp; <math style="vertical-align: -17px">g(x)=\frac{x+3}{x^{\frac{3}{2}}+2}</math> where <math style="vertical-align: 0px">x>0</math>
  
−
<span class="exam">Determine the position and velocity of the object when <math>t=\frac{\pi}{8}.</math>
+
<span class="exam">(c) &nbsp; <math style="vertical-align: -20px">h(x)=\frac{e^{-5x^3}}{\sqrt{x^2+1}}</math>

Latest revision as of 06:58, 3 November 2017

This is a sample, and is meant to represent the material usually covered in Math 9A for the midterm. An actual test may or may not be similar.

Click on the  boxed problem numbers  to go to a solution.

 Problem 1 

Find the following limits:

(a) Find  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim _{x\rightarrow 2} g(x),}   provided that  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim _{x\rightarrow 2} \bigg[\frac{4-g(x)}{x}\bigg]=5.}

(b) Find  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim _{x\rightarrow 0} \frac{\sin(4x)}{5x} }

(c) Evaluate  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim _{x\rightarrow -3^+} \frac{x}{x^2-9} }

 Problem 2 

Suppose the size of a population at time  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t}   is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N(t)=\frac{1000t}{5+t},~t\ge 0.}

(a) Determine the size of the population as  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t\rightarrow \infty.}   We call this the limiting population size.

(b) Show that at time  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=5,}   the size of the population is half its limiting size.

 Problem 3 

Consider the following function  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \left\{ \begin{array}{lr} x^2 & \text{if }x < 1\\ \sqrt{x} & \text{if }x \geq 1 \end{array} \right. }

(a) Find  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\rightarrow 1^-} f(x).}

(b) Find  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\rightarrow 1^+} f(x).}

(c) Find  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\rightarrow 1} f(x).}

(d) Is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f}   continuous at  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=1?}   Briefly explain.

 Problem 4 

Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=\sqrt{3x-5}.}

(a) Use the definition of the derivative to compute   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dx}}   for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=\sqrt{3x-5}.}

(b) Find the equation of the tangent line to  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=\sqrt{3x-5}}   at  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2,1).}

 Problem 5 

Find the derivatives of the following functions. Do not simplify.

(a)   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\sqrt{x}(x^2+2)}

(b)   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x)=\frac{x+3}{x^{\frac{3}{2}}+2}} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x>0}

(c)   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(x)=\frac{e^{-5x^3}}{\sqrt{x^2+1}}}

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