Difference between revisions of "007A Sample Midterm 2"

Latest revision as of 12:10, 2 November 2017

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.

Problem 1

Evaluate 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} \frac{\sqrt{x^2+12}-4}{x-2}}

(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(3x)}{\sin(7x)} }

(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 0} x^2\cos\bigg(\frac{1}{x}\bigg) }

Problem 2

Use the definition of the derivative to 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 \frac{dy}{dx}} for the 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 y=\frac{1+x}{3x}.}

Problem 3

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)=x^3(x^{\frac{4}{3}}-1)}

(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 f(x)=\frac{x^3+x^{-3}}{1+6x}} 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 f(x)=\sqrt{3x^2+5x-7}} 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}

Problem 4

Assume 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)} denotes 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} and 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 N(t)} satisfies the equation:

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{dN}{dt}=3N\bigg(1-\frac{N}{20}\bigg).}

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 f(N)=3N\bigg(1-\frac{N}{20}\bigg),~N\ge 0.} Graph 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(N)} as a function of 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} and identify all equilibria. That is, all points 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 \frac{dN}{dt}=0.}

Problem 5

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 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: -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>]] ==
-<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"> Find the derivatives of the following functions. '''Do not simplify.'''
-<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>
+<span class="exam">(c) &nbsp; <math style="vertical-align: -5px">f(x)=\sqrt{3x^2+5x-7}</math>&nbsp; where &nbsp;<math style="vertical-align: 0px">x>0</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 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:
-<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 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>]] ==
-<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?
-<span class="exam">(a) &nbsp; <math style="vertical-align: -5px">f(x)=\tan^3(7x^2+5) </math>
-<span class="exam">(b) &nbsp; <math style="vertical-align: -5px">g(x)=\sin(\cos(e^x)) </math>
-<span class="exam">(c) &nbsp; <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]]'''