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.

    2. MATH
      MATH

  2. Let MATH
    1. Find $f\,^{\prime }(x).$

      2. MATH
       
    2. Find MATH

      3. MATH
  3. Find (or state) $f^{-1}(x).$
    1. $f(x)=3x-4$

      2. MATH
       
    2. $f(x)=\log _{7}x$
        3. MATH
    3. $f(x)=\arctan x$
        4. MATH

  4. Find MATH

    5. MATH
     
  5. 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 \underline{do not evaluate it} .

    6. MATH

    MATH
     

  6. Differentiate each of the following.
    1. MATH

      2. MATH
       
    2. MATH

      3. MATH

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

    8. MATH
    1. 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 $x.$

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

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

      3. MATH
       
    3. Simplify the following as much as possible.

      4. MATH

      MATH

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