1. Let MATH  Find the Taylor series representation for this function centered around $a=\dfrac{\pi }{4}.$Make sure to show at least 4 nonzero terms in your answer.



 

 

 

 

 

 

 

 

 

  1. Let MATH

a.       Find $f\,^{\prime }(x).$



 

 

 

 

 

 


 

b.      Find MATH



 

 

 

 

 

 

 

 

 

  1. Find (or state) $f^{-1}(x).$

 .         

a.       $f(x)=3x-4$

 

 

 

 

 

b.      $f(x)=\log _{7}x$

 

 

 

 

 

c.       $f(x)=\arctan x$

 

 

 

 

  1. Find MATH



 

 

 

 

 

 

 

 

 

 

 

 


 

  1. Use an appropriate trigonometric substitution to change the following integral into a trigonometric integral (i.e., an integral containing only trigonometric functions and only the variable $\theta $). Simplify the trigonometric integral as much as possible, but do not evaluate it .


MATH

 

 

 

 

 

 

 

 


 

  1. Differentiate each of the following.

a.       MATH

 


 

b.      MATH



 

 

  1. a.  Let $f(x)=x^{4}+2x+3.$ Find MATH (That is, find $g\,^{\prime }(3)$ where $g=f^{-1}.$)



 

 

 

 

 

 

 

 

b.  Suppose that Tony has been working on a trigonometric substitution problem. He has used the substitution $x=5\sin \theta .$ His answer in terms of $\theta $ comes out to be MATH Finish the problem by putting the answer back in terms of $x.$



 

 

 

 

 

 

  1. Quick Answers.
    1. What is the exact value of MATH

 

 

 

 


 

    1. If $f(x)$ is the function drawn below, sketch $f^{-1}(x)$ on the same axes.



 

    1. Simplify the following as much as possible.


MATH