Review for exam 2

We have our second exam this coming Friday, September 17th. The problems on the exam will be very much like the problems you see here. I will go over this problem sheet in class on Wednesday but it will help you immensely to think about it on your own first so please work it out to the best of your ability prior to meeting on Wednesday.

Problems

  1. Use the differentiation rules to compute the derivatives of the following functions.

    1. \(\displaystyle f(x) = x^2 \, 2^x\)
    2. \(\displaystyle f(x) = x^3 e^{2x}\)
    3. \(\displaystyle f(x) = \dfrac{\ln(x)}{x^2}\)
    4. \(\displaystyle f(x) = (x^2 + 1)\sin(x)\)
    5. \(\displaystyle f(x) = e^{x^2 + 3x}\)
    6. \(\displaystyle f(x) = \tan(x)\cos(x^2)\)
    7. \(\displaystyle f(x) = \dfrac{x^2 e^x}{1 + x^3}\)
    8. \(\displaystyle f(x) = \ln(\sin(x))\)
    9. \(\displaystyle f(x) = x^5 \tan(3x)\)
    10. \(\displaystyle f(x) = e^{\sin(x^2)}\)
    11. \(\displaystyle f(x) = \dfrac{\sqrt{x} + e^x}{x^2 + 1}\)
    12. \(\displaystyle f(x) = x^2 e^{2x} \sin(\pi x) \cos(x)\)
    13. \(\displaystyle f(x) = \arcsin(3x) + \arctan(x^3)\)

  2. Use logarithmic differentiation to evaluate the derivatives of the following functions:

    1. \(\displaystyle f(x) = x^2 e^{2x} \sin(\pi x) \cos(x)\)
    2. \(\displaystyle f(x) = \frac{e^x \sqrt{x^2+1}}{\sin(x)\sqrt{x^2-1}}\)

  3. Suppose I invest $10,000 at an annual rate of \(5\%\) compounded 4 times per year.

    1. How much will I have after 5 years?
    2. How long will it take my money to double?
      Repeat the problem assuming your money is compounded continuously.

  4. Sketch the graph of \[f(x) = 3\sin\left(\frac{\pi}{2} x\right).\]
    Be sure to clearly indicate the \(x\) and \(y\) intercepts, as well as the maximum and minimum values.

  5. Use the fact that \(\displaystyle \lim_{\theta\to0} \sin(\theta)/\theta = 1\) to compute

    1. \(\displaystyle \lim_{x\to0}\frac{\sin(15x)}{4x}\)
    2. \(\displaystyle \lim_{h\to0}\frac{\cos(h)-1}{h}\)

  6. Supposing that \(f\) and \(g\) are differentiable functions, use the definition of the derivative to show that

    1. \(\displaystyle \frac{d}{dx} f(x)+3g(x) = f'(x)+3g'(x)\)
    2. \(\displaystyle \frac{d}{dx} x^2\,f(x) = 2x\,f(x)+x^2\,f'(x)\)

  7. The graph of the equation \(x^{4}+y^{2}-3xy=1\) is shown in Figure 1.

    1. Explain how you know for sure that the point \((1,0)\) is on the graph.
    2. Use implicit differentiation express \(y'\) as a function in terms of both \(x\) and \(y\).
    3. Find an equation of the line that’s tangent to the graph at the point \((1,0)\).

  8. The graph of \(f(x)=x^{3}+\frac{1}{2}x-\frac{3}{2}\), together with the line \(y=x\), is shown in Figure 2.

    1. Sketch the graph of of \(f^{-1}(x)\) right on the same set of axes.
    2. Evaluate \((f^{-1})'(0)\).

Figures

Figure 1: The graph of \(x^{4}+y^{2}-3xy=1\)
Figure 2: The graph of \(f(x)=x^{3}+\frac{1}{2}x-\frac{3}{2}\)

Questions

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