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Finding the tangent line at pi/3 for the following equation

  1. Jan 10, 2014 #1
    Hello All,

    I have been trying to brush up on some calculus (differential, I haven't learned integral yet) on my own by finding whatever calculus problems I can find. Recently I found this question listed as a CLEP practice question, and have been having some difficulty with it. Here is the question:

    1. The problem statement, all variables and given/known data
    "What is the slope of the line tangent to the graph of the function [itex]f(x)=\ln (\sin^2 x+3)[/itex] at the point where [itex]x=\frac{\pi}{3}[/itex]?"

    The correct answer is given as [itex]\frac{2\sqrt{3}}{15}[/itex]


    2. Relevant equations

    The relevant equation is given above.


    3. The attempt at a solution

    No matter how many times I calculate it out, I keep getting [itex]f'(\frac{\pi}{3})=\frac{45\sqrt{3}}{8\pi}[/itex]. Here is the work I have done. The error could be anywhere; in the algebra, trig, or calc methodology (math has never been my strongest subject.)

    [tex]f(x)=\ln ((\sin x)^2+3)[/tex]
    [tex]f'(x)=\frac{1}{x} ((\sin x)^2+3)\cdot 2\sin x\cdot \cos x[/tex]
    [tex]f'(x)=\frac{((\sin x)^2+3)}{x} \cdot 2\sin x\cdot \cos x[/tex]
    [tex]f'(\frac{\pi}{3})=\frac{((\sin \frac{\pi}{3})^2+3)}{\frac{\pi}{3}} \cdot 2\sin \frac{\pi}{3}\cdot \cos \frac{\pi}{3}[/tex]
    [tex]f'(\frac{\pi}{3})=\frac{((\frac{\sqrt{3}}{2})^2+3)}{\frac{\pi}{3}} \cdot 2\frac{\sqrt{3}}{2}\cdot \frac{1}{2}[/tex]
    [tex]f'(\frac{\pi}{3})=(\frac{3}{4}+3) \cdot \frac{3}{\pi} \cdot \frac{2\sqrt{3}}{4}[/tex]
    [tex]f'(\frac{\pi}{3})=(\frac{3}{4}+\frac{12}{4}) \cdot \frac{3}{\pi} \cdot \frac{2\sqrt{3}}{4}[/tex]
    [tex]f'(\frac{\pi}{3})=\frac{15}{4} \cdot \frac{3}{\pi} \cdot \frac{2\sqrt{3}}{4}[/tex]
    [tex]f'(\frac{\pi}{3})=\frac{15}{4} \cdot \frac{3}{\pi} \cdot \frac{\sqrt{3}}{2}[/tex]
    [tex]f'(\frac{\pi}{3})=\frac{45}{4\pi} \cdot \frac{\sqrt{3}}{2}[/tex]
    [tex]f'(\frac{\pi}{3})=\frac{45\sqrt{3}}{8\pi}[/tex]
     
    Last edited: Jan 10, 2014
  2. jcsd
  3. Jan 10, 2014 #2

    Student100

    User Avatar
    Education Advisor
    Gold Member

    Check how you're calculating that step; perhaps you would do better to use a u sub here.

    u = (sin(x))^2 + 3)
     
  4. Jan 10, 2014 #3
    Aha! Thank you. So from there, I get:

    [tex]f'(x) = \frac {1}{\sin^2x+3} \cdot 2 \sin x \cdot \cos x[/tex]
    [tex]f'(\frac{\pi}{3}) = \frac {2 \cdot \frac {\sqrt3} {2} \cdot \frac {1}{2}}{\frac{3}{4}+\frac{3}{1}}[/tex]
    [tex]f'(\frac{\pi}{3}) = \frac {\frac {\sqrt3} {2}}{\frac{3}{4}+\frac{12}{4}}[/tex]
    [tex]f'(\frac{\pi}{3}) = \frac {\frac {\sqrt3} {2}}{\frac{15}{4}}[/tex]
    [tex]f'(\frac{\pi}{3}) = \frac {\sqrt3} {2} \cdot \frac{4}{15}[/tex]
    [tex]f'(\frac{\pi}{3}) = \frac {2\sqrt3}{15}[/tex]
     
  5. Jan 10, 2014 #4

    Mark44

    Staff: Mentor

    You can save yourself some writing by not starting each equation with f'(∏/3). As you simplify each expression, just connect the new expression with =.
    $$f'(\frac{\pi}{3}) = \frac {2 \cdot \frac {\sqrt3} {2} \cdot \frac {1}{2}}{\frac{3}{4}+\frac{3}{1}} = \frac {\frac {\sqrt3} {2}}{\frac{3}{4}+\frac{12}{4}}$$
    $$ = \frac {\frac {\sqrt3} {2}}{\frac{15}{4}}$$
    and so on.
     
  6. Jan 17, 2014 #5
    Ok, I'll remember that.
     
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