Difference between revisions of "007A Sample Midterm 1"

Latest revision as of 12:50, 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

Find 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} g(x),} provided 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 \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=2x^2-3x+1.}

(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}.}

(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=2x^2-3x+1} 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 (2,3).}

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)=\sqrt{x+\sqrt{x}}}

Problem 5

To determine drug dosages, doctors estimate a person's body surface area (BSA) (in meters squared) using the formula:

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 \text{BSA}=\frac{\sqrt{hm}}{60}}

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 h} is the height in centimeters 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 m} is the mass in kilograms. Calculate the rate of change of BSA with respect to height for a person of a constant mass 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 m=85.} What is the rate 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 h=170} and $h=190?$ Express your results in the correct units. Does the BSA increase more rapidly with respect to height at lower or higher heights?

Contributions to this page were made by Kayla Murray

@@ Line 39: / Line 39: @@
 == [[007A_Sample Midterm 1,_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"> Find the derivatives of the following functions. '''Do not simplify.'''
 <span class="exam">(a) &nbsp; <math style="vertical-align: -5px">f(x)=\sqrt{x}(x^2+2)</math>
@@ Line 45: / Line 45: @@
 <span class="exam">(b) &nbsp; <math style="vertical-align: -17px">g(x)=\frac{x+3}{x^{\frac{3}{2}}+2}</math> where <math style="vertical-align: 0px">x>0</math>
-<span class="exam">(c) &nbsp; <math style="vertical-align: -20px">h(x)=\frac{e^{-5x^3}}{\sqrt{x^2+1}}</math>
+<span class="exam">(c) &nbsp; <math style="vertical-align: -8px">h(x)=\sqrt{x+\sqrt{x}}</math>
 == [[007A_Sample Midterm 1,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
-<span class="exam"> The displacement from equilibrium of an object in harmonic motion on the end of a spring is:
+<span class="exam"> To determine drug dosages, doctors estimate a person's body surface area (BSA) (in meters squared) using the formula:
-::<span class="exam"><math>y=\frac{1}{3}\cos(12t)-\frac{1}{4}\sin(12t)</math>
+::<span class="exam"><math>\text{BSA}=\frac{\sqrt{hm}}{60}</math>
+<span class="exam">where &nbsp;<math style="vertical-align: 0px">h</math>&nbsp; is the height in centimeters and &nbsp;<math style="vertical-align: 0px">m</math>&nbsp; is the mass in kilograms. Calculate the rate of change of BSA with respect to height for a person of a constant mass of &nbsp;<math style="vertical-align: 0px">m=85.</math>&nbsp; What is the rate at &nbsp;<math style="vertical-align: -1px">h=170</math>&nbsp; and &nbsp;<math style="vertical-align: -1px">h=190?</math>&nbsp; Express your results in the correct units. Does the BSA increase more rapidly with respect to height at lower or higher heights?
-<span class="exam">where &nbsp;<math style="vertical-align: -4px">y</math>&nbsp; is measured in feet and &nbsp;<math style="vertical-align: 0px">t</math>&nbsp; is the time in seconds.
-<span class="exam">Determine the position and velocity of the object when &nbsp;<math style="vertical-align: -14px">t=\frac{\pi}{8}.</math>
 '''Contributions to this page were made by [[Contributors|Kayla Murray]]'''