MathAdmin: /* Problem 5 */

2017-05-25T19:11:57Z

Problem 5

MathAdmin: Created page with "'''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''' '''
2017-04-10T16:19:02Z

Created page with "'''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..."

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'''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.'''
<div class="noautonum">TOC</div>

== [[009A_Sample Final 2,_Problem_1| Problem 1 ]] ==
Compute

(a)  <math style="vertical-align: -15px">\lim_{x\rightarrow 4} \frac{\sqrt{x+5}-3}{x-4}</math>

(b)  <math style="vertical-align: -15px">\lim_{x\rightarrow 0} \frac{\sin^2x}{3x}</math>

(c)  <math style="vertical-align: -15px">\lim_{x\rightarrow -\infty} \frac{\sqrt{x^2+2}}{2x-1}</math>

== [[009A_Sample Final 2,_Problem_2| Problem 2 ]] ==
 Let

::<math>f(x) = \left\{
\begin{array}{lr}
\frac{x^2-2x-3}{x-3} & \text{if }x \ne 3\\
5 & \text{if }x = 3
\end{array}
\right.
</math>
 For what values of  <math style="vertical-align: 0px">x</math>  is  <math style="vertical-align: -4px">f</math>  continuous?

== [[009A_Sample Final 2,_Problem_3| Problem 3 ]] ==
Compute   <math>\frac{dy}{dx}.</math>

(a)  <math style="vertical-align: -15px">y=\bigg(\frac{x^2+3}{x^2-1}\bigg)^3</math>

(b)  <math style="vertical-align: -4px">y=x\cos(\sqrt{x+1})</math>

(c)  <math style="vertical-align: -5px">y=\sin^{-1} x</math>

== [[009A_Sample Final 2,_Problem_4| Problem 4 ]] ==
Use implicit differentiation to find an equation of the tangent line to the curve at the given point.

::<math style="vertical-align: -4px">3x^2+xy+y^2=5</math>  at the point  <math style="vertical-align: -5px">(1,-2)</math>

== [[009A_Sample Final 2,_Problem_5| Problem 5 ]] ==
 A lighthouse is located on a small island 3km away from the nearest point  <math style="vertical-align: 0px">P</math>  on a straight shoreline and its light makes 4 revolutions per minute. How fast is the beam of light moving along the shoreline on a point 1km away from  <math style="vertical-align: 0px">P?</math>

== [[009A_Sample Final 2,_Problem_6| Problem 6 ]] ==
 Find the absolute maximum and absolute minimum values of the function

::<math>f(x)=\frac{1-x}{1+x}</math>

on the interval  <math style="vertical-align: -5px">[0,2].</math>

== [[009A_Sample Final 2,_Problem_7| Problem 7 ]] ==

 Show that the equation  <math style="vertical-align: -2px">x^3+2x-2=0</math>  has exactly one real root.

== [[009A_Sample Final 2,_Problem_8| Problem 8 ]] ==

Compute

(a)  <math style="vertical-align: -18px">\lim_{x\rightarrow \infty} \frac{x^{-1}+x}{1+\sqrt{1+x}}</math>

(b)  <math style="vertical-align: -15px">\lim_{x\rightarrow 0} \frac{\sin x}{\cos x-1}</math>

(c)  <math style="vertical-align: -15px">\lim_{x\rightarrow 1} \frac{x^3-1}{x^{10}-1}</math>

== [[009A_Sample Final 2,_Problem_9| Problem 9 ]] ==

A plane begins its takeoff at 2:00pm on a 2500-mile flight. After 5.5 hours, the plane arrives at its destination. Give a precise mathematical reason using the mean value theorem to explain why there are at least two times during the flight when the speed of the plane is 400 miles per hour.

== [[009A_Sample Final 2,_Problem_10| Problem 10 ]] ==

Let
::<math>f(x)=\frac{4x}{x^2+1}</math>

(a) Find all local maximum and local minimum values of  <math style="vertical-align: -4px">f,</math>  find all intervals where  <math style="vertical-align: -4px">f</math>  is increasing and all intervals where  <math style="vertical-align: -4px">f</math>  is decreasing.

(b) Find all inflection points of the function  <math style="vertical-align: -4px">f,</math>  find all intervals where the function  <math style="vertical-align: -4px">f</math>  is concave upward and all intervals where  <math style="vertical-align: -4px">f</math>  is concave downward.

(c) Find all horizontal asymptotes of the graph  <math style="vertical-align: -5px">y=f(x).</math>

(d) Sketch the graph of  <math style="vertical-align: -5px">y=f(x).</math>

@@ Line 40: / Line 40: @@
 == [[009A_Sample Final 2,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
-<span class="exam"> A lighthouse is located on a small island 3km away from the nearest point &nbsp;<math style="vertical-align: 0px">P</math>&nbsp; on a straight shoreline and its light makes 4 revolutions per minute. How fast is the beam of light moving along the shoreline on a point 1km away from &nbsp;<math style="vertical-align: 0px">P?</math>
+<span class="exam"> A lighthouse is located on a small island 3 km away from the nearest point &nbsp;<math style="vertical-align: 0px">P</math>&nbsp; on a straight shoreline and its light makes 4 revolutions per minute. How fast is the beam of light moving along the shoreline on a point 1km away from &nbsp;<math style="vertical-align: 0px">P?</math>
 == [[009A_Sample Final 2,_Problem_6|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 6&nbsp;</span>]] ==