Difference between revisions of "009A Sample Final 1, Problem 4"

From Grad Wiki
Jump to navigation Jump to search
 
(15 intermediate revisions by 2 users not shown)
Line 1: Line 1:
−
<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">You may leave your answers in point-slope form.
  
−
<span class="exam">compute <math style="vertical-align: -12px">\frac{dy}{dx}</math> 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.
+
<hr>
 +
[[009A Sample Final 1, Problem 4 Solution|'''<u>Solution</u>''']]
  
−
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
−
!Foundations: &nbsp;
 
−
|-
 
−
|
 
−
|}
 
  
−
'''Solution:'''
+
[[009A Sample Final 1, Problem 4 Detailed Solution|'''<u>Detailed Solution</u>''']]
  
−
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
−
!Step 1: &nbsp;
 
−
|-
 
−
|First, we compute <math>\frac{dy}{dx}</math>. We get
 
−
|-
 
−
|<math>\frac{dy}{dx}=2x-\sin(\pi(x^2+1))(2\pi x)</math>.
 
−
|}
 
  
−
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
−
!Step 2: &nbsp;
 
−
|-
 
−
|To find the equation of the tangent line, we first find the slope of the line.
 
−
|-
 
−
|Using <math>x_0=1</math> in the formula for <math>\frac{dy}{dx}</math> from Step 1, we get
 
−
|-
 
−
|<math>m=2(1)-\sin(2\pi)2\pi=2</math>.
 
−
|-
 
−
|To get a point on the line, we plug in <math>x_0=1</math> into the equation given.
 
−
|-
 
−
|So, we have <math>y=1^2+\cos(2\pi)=2</math>.
 
−
|-
 
−
|Thus, the equation of the tangent line is <math>y=2(x-1)+2</math>.
 
−
|}
 
−
 
−
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
−
!Final Answer: &nbsp;
 
−
|-
 
−
|<math>\frac{dy}{dx}=2x-\sin(\pi(x^2+1))(2\pi x)</math>
 
−
|-
 
−
|<math>y=2(x-1)+2</math>
 
−
|}
 
 
[[009A_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 15:46, 2 December 2017

If   compute    and find the equation for the tangent line at  

You may leave your answers in point-slope form.

Solution


Detailed Solution


Return to Sample Exam

Retrieved from "https://gradwiki.math.ucr.edu/index.php?title=009A_Sample_Final_1,_Problem_4&oldid=4279"