Spring 16 Final

Problems

  1. Determine the derivatives of each of the following functions:

    1. \(\displaystyle f(x) = x^2 \sin(4x) + 4\)
    2. \(\displaystyle g(x) = \frac{e^{3x} + 1}{1 + 5x} + 2\)
    3. \(\displaystyle h(x) = x^3 \ln\left( 4 + x \cos(2x + 1) \right)\)
    4. \(\displaystyle p(t) = \frac{1}{t} \cdot \frac{1 + \sqrt{t}}{5} \cdot (1 + t)^4\)
    5. \(\displaystyle s(t) = \frac{\sin(3t) \tan(4t)}{t}\)
    6. \(\displaystyle q(t) = t^2 \ln \left( t^2 + 1 \right)\)

  2. Determine the anti-derivative represented by each of the following indefinite integrals.

    1. \(\displaystyle \int \left( e^{x/2} + \sin(5x) \right) dx\)
    2. \(\displaystyle \int \left( 2x - \frac{4}{x} + 1 \right) dx\)
    3. \(\displaystyle \int \frac{\ln(x)}{x} dx\)

  3. A timer will be constructed using a pendulum. The period in seconds, T, for a pendulum of length L meters is

    \[\displaystyle T = 2\pi\sqrt{L/g},\]

    where \(\displaystyle g\) is 9.81 m/sec. The error in the measurement of the period, \(\displaystyle \Delta T\), should be \(\displaystyle \pm 0.05\) seconds when the length is 0.2 m.

    1. Determine the exact resulting error, \(\displaystyle \Delta L\), necessary in the measurement of the length to obtain the indicated error in the period.
    2. Use the linearization of the period in the formula above to \textbf{estimate} the error, \(\displaystyle \Delta L\), necessary in the measurement of the length to obtain the indicated error in the period.

  4. A function, \(\displaystyle f(x)\), is concave down and increases for \(\displaystyle 0 < x < 2\). It is concave down and decreases for \(\displaystyle 2 < x < 4\). It is concave up and decreases for \(\displaystyle 4 < x < 6\). Make a sketch of a function that satisfies this criteria. On a separate set of axes below it make a sketch of the function’s derivative.

  5. Determine each of the following limits.

    1. \(\displaystyle \lim_{x \to 0} (1 + x)^{3/x}\).
    2. \(\displaystyle \lim_{x \to \infty} \frac{\cos(2x) - 1}{x^2}\).

  6. The graph of a function is shown below. Approximate the area under the curve from \(\displaystyle x = 1\) to \(\displaystyle x = 3\) using a Riemann sum with three intervals. Add a sketch of the rectangles to the plot that correspond to the Riemann sum. Show all of your work. (You do not have to evaluate your result and can leave it as a sum, but it must be in a form that can be directly entered into a calculator.)

    Sketch of a Function

    (A graph labeled "Sketch of a Function" with axes labeled \(\displaystyle f(x)\) and \(\displaystyle x\), showing a curve.)

    IMAGE

    Graph of a function on coordinate axes

  7. Use the definition of the derivative to show that

    \[\displaystyle \frac{d}{dx} \left( x + \frac{1}{x} + 2 \right) = 1 - \frac{1}{x^2}.\]

  8. The velocity of an object is given by

    \[\displaystyle v(t) = \sin(2t) + e^{-t}.\]

    The initial position is \(\displaystyle x(0) = 2\)m. Determine the equation for the position at any time.

  9. A container holds 50 liters of water. A valve is closed and is slowly turned. The water is drained from the container in 2 minutes.

    For each statement indicate if it must be true, must be false, or if it is not possible to determine indicate that you cannot tell from the given information. For each statement provide a complete, one sentence explanation for your reasoning.

    1. True/False/Cannot Tell** The container held 25 liters of water after one minute.
    2. True/False/Cannot Tell** At some point in time the rate of change of the volume of water in the container was 25 l/min.
    3. True/False/Cannot Tell** At some point in time the rate of change of the volume of water in the container was 20 l/min.

  10. A rectangular beam will be cut from a cylindrical log whose radius is 15cm. The stiffness of the resulting beam is proportional to the width multiplied by the cube of its depth. Determine the width and depth that will result in the beam with the greatest stiffness.