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Section Practice

These are practice problems for the tests. Solutions are in the next section. While we will not present these, I am happy to answer questions about them in class.

  1. Graph these functions and their derivatives.

    1. \(F(x)=-x^{2}+5\)

    2. \(h(x) = x^3 - 9x\)

    3. \(f(x)=\cos(2x)\)

    4. \(g(t)=|2+3t|\)

  2. Compute the derivative of each function.

    1. \(\dsp g(t)=\sqrt{3t}+\frac{3}{t}-\frac{5}{t^{3}}\)

    2. \(z(t) = (x^2+1)(x^3+2x)\)

    3. \(\dsp y = \frac{2x^3+x}{2-x^2}\)

    4. \(g(t)=(2t^{2}-5t^{8}+4t)^{12}\)

    5. \(F(z)=\sqrt[3]{2z+7z^{3}}\)

  3. Compute and simplify the derivatives of these exponential and logarithmic functions.

    1. \(a(x) = 12-e^{x^2}\)

    2. \(b(x) = \ln(x^2)\)

    3. \(c(x) = e^{\sqrt{x^2-1}}\)

    4. \(d(x) = x\ln(x^3)\)

    5. \(\dsp g(t)=\frac{t+e^{t}}{3t^{2}-10t+5}\)

    6. \(\dsp f(t) = \frac{1+t}{e^{t}}\)

    7. \(n(x) = \ln(xe^{x^2})\)

  4. Compute and simplify the derivatives of these trigonometric functions.

    1. \(C(\beta )=3\beta \cos(5\beta)\)

    2. \(H(x)=(5x^{2}+x)(4x^{4}+\tan(x))\)

    3. \(o(x) = \csc(x)\cot(x)\)

    4. \(p(x) = \sin(\cos(x))\)

    5. \(r(x) = \left(\sin(x)/x \right)^3\)

    6. \(i(x) = \sin^3(x)/x^3\)

  5. Compute and simplify the derivatives of these mixed functions.

    1. \(g(t) = t\sin(t)e^t\)

    2. \(h(x) = \tan(x^3-3x)\)

    3. \(k(t) = \invcos(x^2+3x)\)

    4. \(m(x) = \invsin(5x)\)

    5. \(\dsp q(t) = e^{\sec(t^2-2t)}\)

    6. \(\dsp s(x) = 3^x + x^3\)

    7. \(\dsp t(y) = 2^{y^2-3y}\)

    8. \(\dsp u(x) = x3^{\sin(x)}\)

    9. \(\dsp G(x)=(x+1)^{\sin(x)}\)

  6. Each hyperbolic trigonometric function is defined. Verify that each stated derivative is correct.

    1. Definition: \(\dsp \sinh(x)=\frac{e^{x}-e^{-x}}{2} \;\;\;\;\;\;\;\;\) Show: \((\sinh(x))'=\cosh(x)\)

    2. Definition: \(\dsp \cosh(x)=\frac{e^{x}+e^{-x}}{2}\;\;\;\;\;\;\;\;\) Show: \((\cosh(x))'=\sinh(x)\)

    3. Definition: \(\dsp \tanh(x)=\frac{\sinh(x)}{\cosh(x)} \;\;\;\;\) Show: \((\tanh(x))'=\sech^{2}(x)\)

    4. Definition: \(\dsp \csch(x)=\frac{1}{\sinh(x)} \;\;\;\;\) Show: \((\csch(x))'=-\csch(x)\coth(x)\)

    5. Definition: \(\dsp \sech(x)=\frac{1}{\cosh(x)} \;\;\;\;\) Show: \((\sech(x))'=-\sech(x)\tanh(x)\)

    6. Definition: \(\dsp \coth(x)=\frac{1}{\tanh(x)} \;\;\;\;\) Show: \((\coth(x))'=-\csch ^{2}(x)\)

  7. Show that \(\invsinh(x)=\ln(x+\sqrt{x^{2}+1}).\)

  8. Show that \(\dsp (\invsinh(x))'=\frac{1}{\sqrt{x^{2}+1}}\text{.}\)

  9. Compute the slope of the tangent line to \(\dsp f(x)=\frac{2x}{2^{x}}\) when \(x=0\text{.}\)

  10. Compute the equation of the tangent line to \(g(x)=x\tan(x)\) when \(x=\frac{\pi}{3}\text{.}\)

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