Fall 15 Final

Problems

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

    1. \(\displaystyle f(x) = 4x + 8x^{0.2}\)
    2. \(\displaystyle g(x) = \frac{x}{1+x}\)
    3. \(\displaystyle h(x) = (x^2 + 1)^2 \cdot \sin(2x)\)
    4. \(\displaystyle u(x) = \frac{1}{\sqrt[3]{x}} + e^{-x^2+1}\)
    5. \(\displaystyle u(x) = \cos(\ln(2 + (x^2-1)^3))\)
    6. \(\displaystyle v(x) = \frac{\tan(x^5+1)}{x+1}\)

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

    1. \(\displaystyle \int x^3 - \sqrt{x} dx\)
    2. \(\displaystyle \int e^{2x} dx\)

  3. Evaluate the derivative

    \[\displaystyle \frac{d}{dx} \int_{0}^{x} t \cdot \sec(5t) dt.\]

  4. Determine the maximum and the minimum values of the function

    \[\displaystyle f(x) = x e^{x - x^2}\]

    where \(\displaystyle 0 \leq x \leq 3\).

  5. Find the equation of the tangent line to

    \[\displaystyle x^2 - y^2 = y,\]

    at the point \(\displaystyle (x, y) = (-\sqrt{2}, 1)\).

  6. A function, \(\displaystyle f(t)\), is shown in the plot below. Determine the value of each of the following quantities. Briefly state the reason for your result in one or two complete sentences.

    \[\displaystyle \begin{array}{cc} & 3 \\ & 2 \\ f(t) & 1 \\ & 0 \\ 0 1 2 3 \\ t \\ \end{array}\]
    1. \(\displaystyle \lim_{t \to 1} f(t) =\)
    2. \(\displaystyle f(1) =\)
    3. \(\displaystyle \lim_{t \to 2} f(t) =\)
    4. \(\displaystyle f(2) =\)
    1. Page 8 of 16 Points deducted: ___

    out of a possible 10 points

    IMAGE

    Graph of the derivative of a function

  7. Approximate the integral

    \[\displaystyle \int_0^2 \cos(\pi x)\ dx\]

    using a Riemann sum with three intervals. The intervals should be equal length and use a left hand sum. (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.)

  8. Use the definition of the derivative to show that

    \[\displaystyle \frac{d}{dx} \left( 3x^2 - \frac{1}{x} \right) = 6x + \frac{1}{x^2}.\]

    (hint: the quotient can be broken into two parts, one for each function.)

  9. A sketch of the derivative of a function, \(\displaystyle f'(x)\), is shown in the plot below. Make a sketch of the original function, \(\displaystyle f(x)\), given that \(\displaystyle f(0) = 0\).

    (Please double check the labels on the plots.)

    A Function And Its Derivative

    [A plot with two graphs, one above the other. The top plot is labeled \(\displaystyle f'(x)\) on the y-axis and shows the derivative of a function. The bottom plot is labeled \(\displaystyle f(x)\) on the y-axis and is blank, for the student to sketch \(\displaystyle f(x)\). The x-axis is labeled \(\displaystyle x\) and runs from 0 to 4.]

    1. Page 11 of 16 Points deducted: _____

    out of a possible 10 points

    IMAGE

    Graph of the derivative of a function

  10. A road has two lanes going north and south, and the lanes are separated by a distance of 0.1 miles. One car, traveling North, is traveling at a constant 80 miles per hour. Another car, traveling South is traveling at constant 70 miles per hour. What is the rate of change of the straight line distance between the two cars when they are approaching one another and the straight line distance between the cars is one mile? What is the rate of change of the straight line distance at the moment when they pass each other?

  11. Sketch a plot of the function

    \[\displaystyle f(x) = \frac{2x}{x^2 - 4}\]

    Label your axes and indicate the values of all \(\displaystyle x\)-intercepts, \(\displaystyle y\)-intercepts, and asymptotes including for \(\displaystyle x \to -\infty\) as well as \(\displaystyle x \to \infty\).

  12. A car starts from rest at a stop light. At the end of 10 seconds its position is 100 meters beyond the light. Three statements are given below. 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 Its final speed is 10 meters per second
    2. True/False/Cannot Tell At some point in time its speed was 10 meters per second.
    3. True/False/Cannot Tell It did not move faster than 10 meters per second at any time.

  13. A cistern for storing water will be constructed. Its shape is a right circular cylinder with radius \(\displaystyle R\) and height \(\displaystyle H\). It must be able to hold \(\displaystyle 1000 \ m^3\) of water. The cost of the materials is related to the surface area. It is \$8 per square meter for the sides and is \$10 per square meter for the bottom. (The top of the cistern is open.) What dimensions for the cistern will minimize the cost of the materials?

    \hrule

    1. Page 15 of 16 Points deducted: _____

    \newline

    out of a possible 10 points