007A Sample Midterm 2 - Revision history

Kayla Murray: /* Problem 4 */

2017-11-02T20:10:55Z

Problem 4

Kayla Murray: /* Problem 1 */

2017-11-02T20:06:59Z

Problem 1

Kayla Murray at 20:06, 2 November 2017

2017-11-02T20:06:11Z

Kayla Murray: /* Problem 1 */

2017-11-02T19:55:17Z

Problem 1

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2017-11-02T19:54:06Z

Created page with "'''This is a sample, and is meant to represent the material usually covered in Math 7A for the midterm. An actual test may or may not be similar.''' '''Click on the''' '''<s..."

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'''This is a sample, and is meant to represent the material usually covered in Math 7A for the midterm. An actual test may or may not be similar.'''

'''Click on the''' ''' boxed problem numbers  to go to a solution.'''
<div class="noautonum">TOC</div>

== [[007A_Sample Midterm 2,_Problem_1| Problem 1 ]] ==
 Evaluate the following limits.

(a) Find  <math style="vertical-align: -14px">\lim _{x\rightarrow 2} \frac{\sqrt{x^2+12}-4}{x-2}</math>

(b) Find  <math style="vertical-align: -19px">\lim _{x\rightarrow 0} \frac{\sin(3x)}{\sin(7x)} </math>

(c) Evaluate  <math style="vertical-align: -20px">\lim _{x\rightarrow (\frac{\pi}{2})^-} \tan(x) </math>

== [[007A_Sample Midterm 2,_Problem_2| Problem 2 ]] ==
The function  <math style="vertical-align: -5px">f(x)=3x^7-8x+2</math>  is a polynomial and therefore continuous everywhere.

(a) State the Intermediate Value Theorem.

(b) Use the Intermediate Value Theorem to show that  <math style="vertical-align: -5px">f(x)</math>  has a zero in the interval  <math style="vertical-align: -5px">[0,1].</math>

== [[007A_Sample Midterm 2,_Problem_3| Problem 3 ]] ==
 Use the definition of the derivative to find   <math>\frac{dy}{dx}</math>   for the function  <math style="vertical-align: -12px">y=\frac{1+x}{3x}.</math>

== [[007A_Sample Midterm 2,_Problem_4| Problem 4 ]] ==
 Find the derivatives of the following functions. Do not simplify.

(a)   <math style="vertical-align: -5px">f(x)=x^3(x^{\frac{4}{3}}-1)</math>

(b)   <math style="vertical-align: -14px">g(x)=\frac{x^3+x^{-3}}{1+6x}</math>  where  <math style="vertical-align: 0px">x>0</math>

== [[007A_Sample Midterm 2,_Problem_5| Problem 5 ]] ==
 Find the derivatives of the following functions. Do not simplify.

(a)   <math style="vertical-align: -5px">f(x)=\tan^3(7x^2+5) </math>

(b)   <math style="vertical-align: -5px">g(x)=\sin(\cos(e^x)) </math>

(c)   <math style="vertical-align: -18px">h(x)=\frac{(5x^2+7x)^3}{\ln(x^2+1)} </math>

'''Contributions to this page were made by [[Contributors|Kayla Murray]]'''

@@ Line 26: / Line 26: @@
 == [[007A_Sample Midterm 2,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
-<span class="exam"> Assume &nbsp;<math>N(t)</math>&nbsp; denotes the size of a population at time &nbsp;<math>t</math>&nbsp; and that &nbsp;<math>N(t)</math>&nbsp; satisfies the equation:
+<span class="exam"> Assume &nbsp;<math style="vertical-align: -5px">N(t)</math>&nbsp; denotes the size of a population at time &nbsp;<math style="vertical-align: 0px">t</math>&nbsp; and that &nbsp;<math style="vertical-align: -5px">N(t)</math>&nbsp; satisfies the equation:
 ::<math>\frac{dN}{dt}=3N\bigg(1-\frac{N}{20}\bigg).</math>
-<span class="exam"> Let &nbsp;<math>f(N)=3N\bigg(1-\frac{N}{20}\bigg),~N\ge 0.</math>&nbsp; Graph &nbsp;<math>f(N)</math>&nbsp; as a function of &nbsp;<math>N</math>&nbsp; and identify all equilibria. That is, all points where &nbsp;<math>\frac{dN}{dt}=0.</math>
+<span class="exam"> Let &nbsp;<math style="vertical-align: -16px">f(N)=3N\bigg(1-\frac{N}{20}\bigg),~N\ge 0.</math>&nbsp; Graph &nbsp;<math style="vertical-align: -5px">f(N)</math>&nbsp; as a function of &nbsp;<math style="vertical-align: 0px">N</math>&nbsp; and identify all equilibria. That is, all points where &nbsp;<math style="vertical-align: -15px">\frac{dN}{dt}=0.</math>
 == [[007A_Sample Midterm 2,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==

@@ Line 11: / Line 11: @@
 <span class="exam">(b) Find &nbsp;<math style="vertical-align: -19px">\lim _{x\rightarrow 0} \frac{\sin(3x)}{\sin(7x)} </math>
-<span class="exam">(c) Evaluate &nbsp;<math style="vertical-align: -20px">\lim _{x\rightarrow 0} x^2\cos\bigg(\frac{1}{x}\bigg) </math>
+<span class="exam">(c) Evaluate &nbsp;<math style="vertical-align: -16px">\lim _{x\rightarrow 0} x^2\cos\bigg(\frac{1}{x}\bigg) </math>
 == [[007A_Sample Midterm 2,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==

@@ Line 14: / Line 14: @@
 == [[007A_Sample Midterm 2,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
-<span class="exam">The function &nbsp;<math style="vertical-align: -5px">f(x)=3x^7-8x+2</math>&nbsp; is a polynomial and therefore continuous everywhere.
+<span class="exam"> Use the definition of the derivative to find &nbsp; <math>\frac{dy}{dx}</math> &nbsp; for the function &nbsp;<math style="vertical-align: -12px">y=\frac{1+x}{3x}.</math>
-<span class="exam">(a) State the Intermediate Value Theorem.
+== [[007A_Sample Midterm 2,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
-<span class="exam">(b) Use the Intermediate Value Theorem to show that &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; has a zero in the interval &nbsp;<math style="vertical-align: -5px">[0,1].</math>
+<span class="exam">(a) &nbsp; <math style="vertical-align: -5px">f(x)=x^3(x^{\frac{4}{3}}-1)</math>
-== [[007A_Sample Midterm 2,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
+<span class="exam">(b) &nbsp; <math style="vertical-align: -14px">f(x)=\frac{x^3+x^{-3}}{1+6x}</math>&nbsp; where &nbsp;<math style="vertical-align: 0px">x>0</math>
-<span class="exam"> Use the definition of the derivative to find &nbsp; <math>\frac{dy}{dx}</math> &nbsp; for the function &nbsp;<math style="vertical-align: -12px">y=\frac{1+x}{3x}.</math>
 == [[007A_Sample Midterm 2,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
-<span class="exam"> Find the derivatives of the following functions. Do not simplify.
+<span class="exam"> Assume &nbsp;<math>N(t)</math>&nbsp; denotes the size of a population at time &nbsp;<math>t</math>&nbsp; and that &nbsp;<math>N(t)</math>&nbsp; satisfies the equation:
-<span class="exam">(a) &nbsp; <math style="vertical-align: -5px">f(x)=x^3(x^{\frac{4}{3}}-1)</math>
+::<math>\frac{dN}{dt}=3N\bigg(1-\frac{N}{20}\bigg).</math>
-<span class="exam">(b) &nbsp; <math style="vertical-align: -14px">g(x)=\frac{x^3+x^{-3}}{1+6x}</math>&nbsp; where &nbsp;<math style="vertical-align: 0px">x>0</math>
+<span class="exam"> Let &nbsp;<math>f(N)=3N\bigg(1-\frac{N}{20}\bigg),~N\ge 0.</math>&nbsp; Graph &nbsp;<math>f(N)</math>&nbsp; as a function of &nbsp;<math>N</math>&nbsp; and identify all equilibria. That is, all points where &nbsp;<math>\frac{dN}{dt}=0.</math>
 == [[007A_Sample Midterm 2,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
-<span class="exam"> Find the derivatives of the following functions. Do not simplify.
+<span class="exam"> A kite 30 (meters) above the ground moves horizontally at a speed of 6 (m/s). At what rate is the length of the string increasing when 50 (meters) of the string has been let out?
 '''Contributions to this page were made by [[Contributors|Kayla Murray]]'''

@@ Line 11: / Line 11: @@
 <span class="exam">(b) Find &nbsp;<math style="vertical-align: -19px">\lim _{x\rightarrow 0} \frac{\sin(3x)}{\sin(7x)} </math>
-<span class="exam">(c) Evaluate &nbsp;<math style="vertical-align: -20px">\lim _{x\rightarrow (\frac{\pi}{2})^-} \tan(x) </math>
+<span class="exam">(c) Evaluate &nbsp;<math style="vertical-align: -20px">\lim _{x\rightarrow 0} x^2\cos\bigg(\frac{1}{x}\bigg) </math>
 == [[007A_Sample Midterm 2,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==