Difference between revisions of "009A Sample Midterm 1"
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== [[009A_Sample Midterm 1,_Problem_2|<span class="biglink"><span style="font-size:80%"> Problem 2 </span>]] == | == [[009A_Sample Midterm 1,_Problem_2|<span class="biglink"><span style="font-size:80%"> Problem 2 </span>]] == | ||
| − | <span class="exam">Consider the following function <math> f:</math> | + | <span class="exam">Consider the following function <math style="vertical-align: -5px"> f:</math> |
::<math>f(x) = \left\{ | ::<math>f(x) = \left\{ | ||
\begin{array}{lr} | \begin{array}{lr} | ||
| Line 23: | Line 23: | ||
</math> | </math> | ||
| − | <span class="exam">(a) Find <math> \lim_{x\rightarrow 1^-} f(x).</math> | + | <span class="exam">(a) Find <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 <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 <math style="vertical-align: -13px"> \lim_{x\rightarrow 1} f(x).</math> |
| − | <span class="exam">(d) Is <math>f</math> continuous at <math>x=1?</math> Briefly explain. | + | <span class="exam">(d) Is <math style="vertical-align: -5px">f</math> continuous at <math style="vertical-align: -1px">x=1?</math> Briefly explain. |
== [[009A_Sample Midterm 1,_Problem_3|<span class="biglink"><span style="font-size:80%"> Problem 3 </span>]] == | == [[009A_Sample Midterm 1,_Problem_3|<span class="biglink"><span style="font-size:80%"> Problem 3 </span>]] == | ||
Revision as of 15:39, 18 February 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 provided that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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
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 3
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 4
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}}}
Problem 5
The displacement from equilibrium of an object in harmonic motion on the end of a spring 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 y=\frac{1}{3}\cos(12t)-\frac{1}{4}\sin(12t)}
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 y} is measured in feet and 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 the time in seconds.
Determine the position and velocity of the object when 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=\frac{\pi}{8}.}