Difference between revisions of "009A Sample Final A"
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<span style="font-size:135%"><font face=Times Roman> | <span style="font-size:135%"><font face=Times Roman> | ||
| − | [[009A_Sample_Final_A,_Problem_1|'''1.''']] Find the following limits:<br> (a) <math style="vertical-align: -45%;">\lim_{x\rightarrow0}\frac{\tan(3x)}{x^{3}}.</math> | + | [[009A_Sample_Final_A,_Problem_1|'''Problem 1.''']] Find the following limits:<br> (a) <math style="vertical-align: -45%;">\lim_{x\rightarrow0}\frac{\tan(3x)}{x^{3}}.</math> |
<br><br> | <br><br> | ||
(b) <math style="vertical-align: -52%;">\lim_{x\rightarrow-\infty}\frac{\sqrt{x^{6}+6x^{2}+2}}{x^{3}+x-1}.</math> | (b) <math style="vertical-align: -52%;">\lim_{x\rightarrow-\infty}\frac{\sqrt{x^{6}+6x^{2}+2}}{x^{3}+x-1}.</math> | ||
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== Derivatives == | == Derivatives == | ||
| − | <span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_2|'''2.''']] Find the derivatives of the following functions: | + | <span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_2|'''Problem 2.''']] Find the derivatives of the following functions: |
<br> | <br> | ||
(a) <math style="vertical-align: -45%;">f(x)=\frac{3x^{2}-5}{x^{3}-9}.</math> | (a) <math style="vertical-align: -45%;">f(x)=\frac{3x^{2}-5}{x^{3}-9}.</math> | ||
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== Continuity and Differentiability == | == Continuity and Differentiability == | ||
| − | <span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_3|'''3.''']] (Version I) Consider the following function: | + | <span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_3|'''Problem 3.''']] (Version I) Consider the following function: |
<math style="vertical-align: -80%;">f(x) = \begin{cases} \sqrt{x}, & \mbox{if }x\geq 1, \\ 4x^{2}+C, & \mbox{if }x<1. \end{cases}</math> | <math style="vertical-align: -80%;">f(x) = \begin{cases} \sqrt{x}, & \mbox{if }x\geq 1, \\ 4x^{2}+C, & \mbox{if }x<1. \end{cases}</math> | ||
<br> | <br> | ||
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(b) With your choice of <math style="vertical-align: -0.1%;">C</math>, is <math>f</math> differentiable at <math style="vertical-align: -3%;">x=1</math>? Use the definition of the derivative to motivate your answer. | (b) With your choice of <math style="vertical-align: -0.1%;">C</math>, is <math>f</math> differentiable at <math style="vertical-align: -3%;">x=1</math>? Use the definition of the derivative to motivate your answer. | ||
<br><br> | <br><br> | ||
| − | 3. (Version II) Consider the following function: | + | [[009A_Sample_Final_A,_Problem_3|'''Problem 3.''']] (Version II) Consider the following function: |
<math style="vertical-align: -80%;">g(x)=\begin{cases} | <math style="vertical-align: -80%;">g(x)=\begin{cases} | ||
\sqrt{x^{2}+3}, & \quad\mbox{if } x\geq1\\ | \sqrt{x^{2}+3}, & \quad\mbox{if } x\geq1\\ | ||
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== Implicit Differentiation == | == Implicit Differentiation == | ||
<span style="font-size:135%"><font face=Times Roman> | <span style="font-size:135%"><font face=Times Roman> | ||
| − | [[009A_Sample_Final_A,_Problem_4 |'''4.''']] Find an equation for the tangent | + | [[009A_Sample_Final_A,_Problem_4 |'''Problem 4.''']] Find an equation for the tangent |
line to the function <math style="vertical-align: -13%;">-x^{3}-2xy+y^{3}=-1</math> at the point <math style="vertical-align: -15%;">(1,1)</math>. </font face=Times Roman> </span> | line to the function <math style="vertical-align: -13%;">-x^{3}-2xy+y^{3}=-1</math> at the point <math style="vertical-align: -15%;">(1,1)</math>. </font face=Times Roman> </span> | ||
== Derivatives and Graphing == | == Derivatives and Graphing == | ||
| − | <span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_5 |'''5.''']] Consider the function | + | <span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_5 |'''Problem 5.''']] Consider the function |
| | ||
<math style="vertical-align: -42%;">h(x)={\displaystyle \frac{x^{3}}{3}-2x^{2}-5x+\frac{35}{3}}.</math> | <math style="vertical-align: -42%;">h(x)={\displaystyle \frac{x^{3}}{3}-2x^{2}-5x+\frac{35}{3}}.</math> | ||
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== Asymptotes == | == Asymptotes == | ||
| − | <br><span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_6 |'''6.''']] Find the vertical and horizontal asymptotes of the function</font face=Times Roman> </span> | + | <br><span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_6 |'''Problem 6.''']] Find the vertical and horizontal asymptotes of the function</font face=Times Roman> </span> |
<math style="vertical-align: -60%;">f(x)=\frac{\sqrt{4x^{2}+3}}{10x-20}.</math> | <math style="vertical-align: -60%;">f(x)=\frac{\sqrt{4x^{2}+3}}{10x-20}.</math> | ||
<br> | <br> | ||
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== Optimization == | == Optimization == | ||
<br> | <br> | ||
| − | <span style="font-size:135%"><font face=Times Roman> [[009A_Sample_Final_A,_Problem_7 |'''7.''']] A farmer wishes to make 4 identical rectangular pens, each with | + | <span style="font-size:135%"><font face=Times Roman> [[009A_Sample_Final_A,_Problem_7 |'''Problem 7.''']] A farmer wishes to make 4 identical rectangular pens, each with |
500 sq. ft. of area. What dimensions for each pen will use the least | 500 sq. ft. of area. What dimensions for each pen will use the least | ||
amount of total fencing? </font face=Times Roman> </span> | amount of total fencing? </font face=Times Roman> </span> | ||
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== Linear Approximation == | == Linear Approximation == | ||
<br> | <br> | ||
| − | <span style="font-size:135%"> <font face=Times Roman>[[009A_Sample_Final_A,_Problem_8|'''8.''']] (a) Find the linear approximation <math style="vertical-align: -14%;">L(x)</math> to the function <math style="vertical-align: -14%;">f(x)=\sec x</math> at the point <math style="vertical-align: -14%;">x=\pi/3</math>. | + | <span style="font-size:135%"> <font face=Times Roman>[[009A_Sample_Final_A,_Problem_8|'''Problem 8.''']] (a) Find the linear approximation <math style="vertical-align: -14%;">L(x)</math> to the function <math style="vertical-align: -14%;">f(x)=\sec x</math> at the point <math style="vertical-align: -14%;">x=\pi/3</math>. |
<br> | <br> | ||
(b) Use <math style="vertical-align: -14%;">L(x)</math> to estimate the value of <math style="vertical-align: -14%;">\sec\,(3\pi/7)</math>. </font face=Times Roman> </span> | (b) Use <math style="vertical-align: -14%;">L(x)</math> to estimate the value of <math style="vertical-align: -14%;">\sec\,(3\pi/7)</math>. </font face=Times Roman> </span> | ||
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== Related Rates == | == Related Rates == | ||
<br> | <br> | ||
| − | <span style="font-size:135%"> <font face=Times Roman> [[009A_Sample_Final_A,_Problem_9|'''9.''']] A bug is crawling along the <math style="vertical-align: 0%;">x</math>-axis at a constant speed of <math style="vertical-align: -42%;">\frac{dx}{dt}=30</math>. | + | <span style="font-size:135%"> <font face=Times Roman> [[009A_Sample_Final_A,_Problem_9|'''Problem 9.''']] A bug is crawling along the <math style="vertical-align: 0%;">x</math>-axis at a constant speed of <math style="vertical-align: -42%;">\frac{dx}{dt}=30</math>. |
How fast is the distance between the bug and the point <math style="vertical-align: -14%;">(3,4)</math> changing | How fast is the distance between the bug and the point <math style="vertical-align: -14%;">(3,4)</math> changing | ||
when the bug is at the origin? ''(Note that if the distance is decreasing, then you should have a negative answer)''. </font face=Times Roman> </span> | when the bug is at the origin? ''(Note that if the distance is decreasing, then you should have a negative answer)''. </font face=Times Roman> </span> | ||
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== Two Important Theorems == | == Two Important Theorems == | ||
| − | <span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_10|'''10.''']] Consider the function | + | <span style="font-size:135%"><font face=Times Roman>[[009A_Sample_Final_A,_Problem_10|'''Problem 10.''']] Consider the function |
| | ||
<math style="vertical-align: -15%;">f(x)=2x^{3}+4x+\sqrt{2}.</math> | <math style="vertical-align: -15%;">f(x)=2x^{3}+4x+\sqrt{2}.</math> | ||
Revision as of 10:27, 30 March 2015
This is a sample final, and is meant to represent the material usually covered in Math 9A. Moreover, it contains enough questions to represent a three hour test. An actual test may or may not be similar. Click on the blue problem numbers to go to a solution.
Limits
Problem 1. Find the following limits:
(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\rightarrow0}\frac{\tan(3x)}{x^{3}}.}
(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-\infty}\frac{\sqrt{x^{6}+6x^{2}+2}}{x^{3}+x-1}.}
(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\rightarrow3}\frac{x-3}{\sqrt{x+1}-2}.}
(d) 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\rightarrow3}\frac{x-1}{\sqrt{x+1}-1}.}
(e) 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{5x^{2}-2x+3}{1-3x^{2}}.}
Derivatives
Problem 2. 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)=\frac{3x^{2}-5}{x^{3}-9}.}
(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)=\pi+2\cos(\sqrt{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 h(x)=4x\sin(x)+e(x^{2}+2)^{2}.}
Continuity and Differentiability
Problem 3. (Version I) 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) = \begin{cases} \sqrt{x}, & \mbox{if }x\geq 1, \\ 4x^{2}+C, & \mbox{if }x<1. \end{cases}}
(a) Find a value 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 C}
which makes 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.}
(b) With your choice 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 C}
, 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}
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=1}
? Use the definition of the derivative to motivate your answer.
Problem 3. (Version II) 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 g(x)=\begin{cases} \sqrt{x^{2}+3}, & \quad\mbox{if } x\geq1\\ \frac{1}{4}x^{2}+C, & \quad\mbox{if }x<1. \end{cases}}
(a) Find a value 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 C}
which makes 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.}
(b) With your choice 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 C}
, 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}
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=1}
? Use the definition of the derivative to motivate your answer.
Implicit Differentiation
Problem 4. Find an equation for the tangent line to 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 -x^{3}-2xy+y^{3}=-1} 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 (1,1)} .
Derivatives and Graphing
Problem 5. Consider 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 h(x)={\displaystyle \frac{x^{3}}{3}-2x^{2}-5x+\frac{35}{3}}.}
(a) Find the intervals where the function is increasing and decreasing.
(b) Find the local maxima and minima.
(c) Find the intervals on which 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 concave upward and concave
downward.
(d) Find all inflection points.
(e) Use the information in the above 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 f(x)}
.
Asymptotes
Problem 6. Find the vertical and horizontal asymptotes of 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)=\frac{\sqrt{4x^{2}+3}}{10x-20}.}
Optimization
Problem 7. A farmer wishes to make 4 identical rectangular pens, each with
500 sq. ft. of area. What dimensions for each pen will use the least
amount of total fencing?
Linear Approximation
Problem 8. (a) Find the linear approximation 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 L(x)}
to 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)=\sec x}
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 x=\pi/3}
.
(b) Use 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 L(x)}
to estimate the value 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 \sec\,(3\pi/7)}
.
Related Rates
Problem 9. A bug is crawling along the 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}
-axis at a constant speed 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 \frac{dx}{dt}=30}
.
How fast is the distance between the bug and 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 (3,4)}
changing
when the bug is at the origin? (Note that if the distance is decreasing, then you should have a negative answer).
Two Important Theorems
Problem 10. Consider 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)=2x^{3}+4x+\sqrt{2}.}
(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 Rolle's 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 exactly one zero.