Prof. A. Suciu

MTH 1733 -- Honors Calculus 1 -- Fall 1997

Quiz 1


In Problems 1 and 2, H is infinite, $\epsilon$ is (a non-zero) infinitesimal, and x is standard.

  1. Tell whether each of the following expressions is infinitesimal, moderate, or infinite.
    (a)
    $\displaystyle{\frac{\epsilon- 2}{\epsilon^2}}$
    (b)
    $\displaystyle{\frac{\epsilon^2 - \displaystyle{\frac{1}{\epsilon}}} {\epsilon^2+\displaystyle{\frac{1}{\epsilon}}}}$

    (c)
    $\displaystyle{\frac{\sqrt{H^2+1}}{3H-2}}-\frac{1}{2}$

    (d)
    $\displaystyle{\frac{5H+6}{H^2+3H+2}}$

  2. Find the standard parts of each of the following numbers.

    (a)
    $\displaystyle{\frac{\displaystyle{\frac{2}{x+\epsilon}-\frac{2}{x}}}{\epsilon}}$
    (b)
    $\displaystyle{\frac{(H+1)^2-5}{H^2+2H+1}}$

    (c)
    $\displaystyle{\frac{27 - (3+\epsilon)^3}{\epsilon}}$

  3. Determine whether the following functions are differentiable at the indicated point (using the definition of the derivative). If differentiable, calculate the derivative at that point.

    (a)
    $f(x) = \displaystyle{\frac{1-\vert x\vert}{1+\vert x\vert}}$ at x=0
    (b)
    f(x) = x|x3 - 2| at x=0

  4. Find the derivatives of the following functions:

    (a)
    $y=\displaystyle{\frac{1}{2}x^4 + \frac{2}{5}}$

    (b)
    $y=\displaystyle{\sqrt{x^3} - \displaystyle{\frac{1}{\sqrt{x}}}}$

  5. Let f(x)=x2-3x+1.

    (a)
    Find the equation of the tangent line to the graph y=f(x) at the point where x=2.

    (b)
    At what point on the graph y=f(x) is the tangent line parallel to the line y=7x-6?

    About this document ...
    Back to MTH 1733, or back to my Home page.

    Created by Alexandru Suciu, Thu Oct 2, 1997
    alexsuciu@neu.edu
    http://www.math.neu.edu/~suciu/mth1733/1733f97.q1/index.html