Skip to main content

In class practice for exam 1

Here are a few more practice problems for our first exam this Friday.

  1. Let’s suppose that we know that \[ \lim_{x\to -1} f(x) = 2 \text{ and } \lim_{x\to-1} g(x) = -3. \] What can you say about
    \[\lim_{x\to-1} (4f(x)-2g(x))?\]

  2. Write down a complete sentence referring to the intermediate value theorem explaining why \[f(x)= \sqrt[3]{x} + \frac{1}{4}x - 2\] has a root between \(x=0\) and \(x=8\).

  3. Find the derivatives of the following functions using the basic differentiation rules.

    1. \(f(x) = 4x^2 - 1\)
    2. \(f(x) = 42x^{18} - 3x^{15} + 5x^{4} - 10\)
    3. \(\displaystyle f(x) = \frac{x^4 - 2x^{5/2} - x + \sqrt{x}}{x^2}\).

  4. Find the derivative of the following function, using the definition of the derivative: \[f(x) = 4x^2 - 1.\]

  5. Let \(f(x) = 5x-x^2\).

    1. Sketch the graph of \(f\), together with the its tangent line at the point \(x=2\).
    2. Find a formula for the tangent line.
  1. The graph of the function \(f\) is shown in the top part of Figure 1 below. It consists of a semi-circle and two line segments. Sketch the graph of \(f'\) on the axes provided below the graph of \(f\).

Piecewise function graph used as the basis for sketching its derivative.

Blank coordinate axes for sketching a derivative.

Figure 1: The graph of a function, together with an empty pair of axes for problem 6