Difference between revisions of "009A Sample Final 1"

Latest revision as of 08:05, 10 April 2017

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

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

Problem 1

In each part, compute the limit. If the limit is infinite, be sure to specify positive or negative infinity.

(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 \lim_{x\rightarrow -3} \frac{x^3-9x}{6+2x}}

(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 \lim_{x\rightarrow 0^+} \frac{\sin (2x)}{x^2}}

(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 \lim_{x\rightarrow -\infty} \frac{3x}{\sqrt{4x^2+x+5}}}

Problem 2

Consider the following piecewise defined 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(x) = \left\{ \begin{array}{lr} x+5 & \text{if }x < 3\\ 4\sqrt{x+1} & \text{if }x \geq 3 \end{array} \right. }

(a) Show 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 f(x)} is 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=3.}

(b) Using the limit definition of the derivative, and computing the limits from both sides, show 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 f(x)} is differentiable 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=3} .

Problem 3

Find the derivatives of the following functions.

(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)=\ln \bigg(\frac{x^2-1}{x^2+1}\bigg)}

(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)=2\sin (4x)+4\tan (\sqrt{1+x^3})}

Problem 4

If 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=\cos^{-1} (2x)} 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}} and find the equation for the tangent line 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_0=\frac{\sqrt{3}}{4}.}

You may leave your answers in point-slope form.

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?

Problem 6

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(x)=3x-2\sin x+7}

(a) Use the Intermediate Value Theorem to show 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 f(x)} has at least one zero.

(b) Use the Mean Value Theorem to show 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 f(x)} has at most one zero.

Problem 7

A curve is defined implicitly by 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 x^3+y^3=6xy.}

(a) Using implicit differentiation, 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}} .

(b) Find an equation of the tangent line to the curve 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^3+y^3=6xy} at the point 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 (3,3)} .

Problem 8

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=x^3.}

(a) Find the differential 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 dy} 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 y=x^3} 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=2} .

(b) Use differentials to find an approximate value 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 1.9^3} .

Problem 9

Given 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 f(x)=x^3-6x^2+5} ,

(a) Find the intervals in which the function increases or decreases.

(b) Find the local maximum and local minimum values.

(c) Find the intervals in which the function concaves upward or concaves downward.

(d) Find the inflection point(s).

(e) Use the above information (a) to (d) to sketch the graph 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 y=f(x)} .

Problem 10

Consider the following continuous 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(x)=x^{1/3}(x-8)}

defined on the closed, bounded interval 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 [-8,8]} .

(a) Find all the critical points 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 f(x)} .

(b) Determine the absolute maximum and absolute minimum values 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 f(x)} on the interval 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 [-8,8]} .

Contributions to this page were made by Kayla Murray

@@ Line 1: / Line 1: @@
-'''This is a sample, and is meant to represent the material usually covered in Math 9A for the final. An actual test may or may not be similar. Click on the''' '''<span class="biglink" style="color:darkblue;">&nbsp;boxed problem numbers&nbsp;</span> to go to a solution.'''
+'''This is a sample, and is meant to represent the material usually covered in Math 9A for the final. An actual test may or may not be similar.'''
+'''Click on the''' '''<span class="biglink" style="color:darkblue;">&nbsp;boxed problem numbers&nbsp;</span> to go to a solution.'''
 <div class="noautonum">__TOC__</div>
@@ Line 5: / Line 7: @@
 <span class="exam">In each part, compute the limit. If the limit is infinite, be sure to specify positive or negative infinity.
-<span class="exam">a) <math style="vertical-align: -14px">\lim_{x\rightarrow -3} \frac{x^3-9x}{6+2x}</math>
+<span class="exam">(a) &nbsp; <math style="vertical-align: -14px">\lim_{x\rightarrow -3} \frac{x^3-9x}{6+2x}</math>
-<span class="exam">b) <math style="vertical-align: -14px">\lim_{x\rightarrow 0^+} \frac{\sin (2x)}{x^2}</math>
+<span class="exam">(b) &nbsp; <math style="vertical-align: -14px">\lim_{x\rightarrow 0^+} \frac{\sin (2x)}{x^2}</math>
-<span class="exam">c) <math style="vertical-align: -14px">\lim_{x\rightarrow -\infty} \frac{3x}{\sqrt{4x^2+x+5}}</math>
+<span class="exam">(c) &nbsp; <math style="vertical-align: -14px">\lim_{x\rightarrow -\infty} \frac{3x}{\sqrt{4x^2+x+5}}</math>
 == [[009A_Sample Final 1,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
 <span class="exam"> Consider the following piecewise defined function:
-::::::<math>f(x) = \left\{
+::<math>f(x) = \left\{
       \begin{array}{lr}
         x+5 &  \text{if }x < 3\\
@@ Line 21: / Line 23: @@
     \right.
 </math>
-<span class="exam">a) Show that <math style="vertical-align: -5px">f(x)</math> is continuous at <math style="vertical-align: 0px">x=3</math>.
+<span class="exam">(a) Show that &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is continuous at &nbsp;<math style="vertical-align: 0px">x=3.</math>
-<span class="exam">b) Using the limit definition of the derivative, and computing the limits from both sides, show that <math style="vertical-align: -5px">f(x)</math> is differentiable at <math style="vertical-align: 0px">x=3</math>.
+<span class="exam">(b) Using the limit definition of the derivative, and computing the limits from both sides, show that &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; is differentiable at &nbsp;<math style="vertical-align: 0px">x=3</math>.
 == [[009A_Sample Final 1,_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.
-<span class="exam">a) <math style="vertical-align: -14px">f(x)=\ln \bigg(\frac{x^2-1}{x^2+1}\bigg)</math>
+<span class="exam">(a) &nbsp; <math style="vertical-align: -14px">f(x)=\ln \bigg(\frac{x^2-1}{x^2+1}\bigg)</math>
-<span class="exam">b) <math style="vertical-align: -3px">g(x)=2\sin (4x)+4\tan (\sqrt{1+x^3})</math>
+<span class="exam">(b) &nbsp; <math style="vertical-align: -3px">g(x)=2\sin (4x)+4\tan (\sqrt{1+x^3})</math>
 == [[009A_Sample Final 1,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
-<span class="exam"> If
+<span class="exam"> If &nbsp;<math style="vertical-align: -5px">y=\cos^{-1} (2x)</math> compute &nbsp;<math style="vertical-align: -12px">\frac{dy}{dx}</math>&nbsp; and find the equation for the tangent line at &nbsp;<math style="vertical-align: -14px">x_0=\frac{\sqrt{3}}{4}.</math>
-::::::<math>y=x^2+\cos (\pi(x^2+1))</math>
-<span class="exam">compute &thinsp;<math style="vertical-align: -12px">\frac{dy}{dx}</math>&thinsp; and find the equation for the tangent line at <math style="vertical-align: -3px">x_0=1</math>. You may leave your answers in point-slope form.
+<span class="exam">You may leave your answers in point-slope form.
 == [[009A_Sample Final 1,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
@@ Line 45: / Line 45: @@
 <span class="exam"> Consider the following function:
-::::::<math>f(x)=3x-2\sin x+7</math>
+::<math>f(x)=3x-2\sin x+7</math>
-<span class="exam">a) Use the Intermediate Value Theorem to show that <math style="vertical-align: -5px">f(x)</math> has at least one zero.
+<span class="exam">(a) Use the Intermediate Value Theorem to show that &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; has at least one zero.
-<span class="exam">b) Use the Mean Value Theorem to show that <math style="vertical-align: -5px">f(x)</math> has at most one zero.
+<span class="exam">(b) Use the Mean Value Theorem to show that &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; has at most one zero.
 == [[009A_Sample Final 1,_Problem_7|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 7&nbsp;</span>]] ==
@@ Line 55: / Line 55: @@
 <span class="exam">A curve is defined implicitly by the equation
-::::::<math>x^3+y^3=6xy.</math>
+::<math>x^3+y^3=6xy.</math>
-<span class="exam">a) Using implicit differentiation, compute &thinsp;<math style="vertical-align: -12px">\frac{dy}{dx}</math>.
+<span class="exam">(a) Using implicit differentiation, compute &nbsp;<math style="vertical-align: -12px">\frac{dy}{dx}</math>.
-<span class="exam">b) Find an equation of the tangent line to the curve <math style="vertical-align: -4px">x^3+y^3=6xy</math> at the point <math style="vertical-align: -5px">(3,3)</math>.
+<span class="exam">(b) Find an equation of the tangent line to the curve &nbsp;<math style="vertical-align: -4px">x^3+y^3=6xy</math>&nbsp; at the point &nbsp;<math style="vertical-align: -5px">(3,3)</math>.
 == [[009A_Sample Final 1,_Problem_8|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 8&nbsp;</span>]] ==
@@ Line 65: / Line 65: @@
 <span class="exam">Let
-::::::<math>y=x^3.</math>
+::<math>y=x^3.</math>
-<span class="exam">a) Find the differential <math style="vertical-align: -4px">dy</math> of <math style="vertical-align: -4px">y=x^3</math> at <math style="vertical-align: 0px">x=2</math>.
+<span class="exam">(a) Find the differential &nbsp;<math style="vertical-align: -4px">dy</math>&nbsp; of &nbsp;<math style="vertical-align: -4px">y=x^3</math>&nbsp; at &nbsp;<math style="vertical-align: 0px">x=2</math>.
-<span class="exam">b) Use differentials to find an approximate value for <math style="vertical-align: -1px">1.9^3</math>.
+<span class="exam">(b) Use differentials to find an approximate value for &nbsp;<math style="vertical-align: -1px">1.9^3</math>.
 == [[009A_Sample Final 1,_Problem_9|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 9&nbsp;</span>]] ==
-<span class="exam">Given the function <math style="vertical-align: -5px">f(x)=x^3-6x^2+5</math>,
+<span class="exam">Given the function &nbsp;<math style="vertical-align: -5px">f(x)=x^3-6x^2+5</math>,
-<span class="exam">a) Find the intervals in which the function increases or decreases.
+<span class="exam">(a) Find the intervals in which the function increases or decreases.
-<span class="exam">b) Find the local maximum and local minimum values.
+<span class="exam">(b) Find the local maximum and local minimum values.
-<span class="exam">c) Find the intervals in which the function concaves upward or concaves downward.
+<span class="exam">(c) Find the intervals in which the function concaves upward or concaves downward.
-<span class="exam">d) Find the inflection point(s).
+<span class="exam">(d) Find the inflection point(s).
-<span class="exam">e) Use the above information (a) to (d) to sketch the graph of <math style="vertical-align: -5px">y=f(x)</math>.
+<span class="exam">(e) Use the above information (a) to (d) to sketch the graph of &nbsp;<math style="vertical-align: -5px">y=f(x)</math>.
 == [[009A_Sample Final 1,_Problem_10|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 10&nbsp;</span>]] ==
 <span class="exam">Consider the following continuous function:
-::::::<math>f(x)=x^{1/3}(x-8)</math>
+::<math>f(x)=x^{1/3}(x-8)</math>
+<span class="exam">defined on the closed, bounded interval &nbsp;<math style="vertical-align: -5px">[-8,8]</math>.
+<span class="exam">(a) Find all the critical points for &nbsp;<math style="vertical-align: -5px">f(x)</math>.
-<span class="exam">defined on the closed, bounded interval <math style="vertical-align: -5px">[-8,8]</math>.
+<span class="exam">(b) Determine the absolute maximum and absolute minimum values for &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; on the interval &nbsp;<math style="vertical-align: -5px">[-8,8]</math>.
-<span class="exam">a) Find all the critical points for <math style="vertical-align: -5px">f(x)</math>.
-<span class="exam">b) Determine the absolute maximum and absolute minimum values for <math style="vertical-align: -5px">f(x)</math> on the interval <math style="vertical-align: -5px">[-8,8]</math>.
+'''Contributions to this page were made by [[Contributors|Kayla Murray]]'''