Math 121 take home

  1. The slope of the tangent line for a function $f$ at $x=a$ is defined to beMATH

    1. If $f(x)=x^{2}-2x,$ use $h=0.001$ and $h=-0.001$ respectively to approximate $f^{^{\prime }}(2).$

    2. If $f(x)=-x^{2}+x,$use $h=0.001$ and $h=-0.001$ respectively to approximate $f^{^{\prime }}(2).$

  2. Find a profit function $P(x)$ satisfies the following conditions, where $x$ denotes number of units are produced and $P(x)$ is a polynomial function: (a) the company breaks even at $x=100,400$ and $600$ units (b) the profit function $P(x)$ changes signs (from $P(x)>0$ to $P(x)<0$ or vise versa) at $x=400$ but $P(x)$ does not change sign at $x=100$ and $600,$ (c) the company is NOT profitable when $50$ units are produced.

  3. Sketch the function above.

  4. Let $f(x)=-\sqrt{-x}.$

    1. Sketch $y=f(x).$

    2. Find the function $f_{1}$ so that the graph of $y=f_{1}(x)$ is being shifted to the left $2$ units and down $3$ units from $f.$

    3. Find the inverse function for $f_{1}(x)$ and graph the inverse function $y=f_{1}^{-1}(x)$ together with $y=f_{1}(x).$

  5. Let $f$ be the function graphed below:
    graphics/takehome__36.png

    1. Does $f$ have an inverse in the domain of $[-2,5]?$ Explain.

    2. If we restrict the domain of $f$ to be $[2,5],$ does $f^{-1}$ exist? Explain.

    3. If $f(2)=1,$ $f(3)=2$ and $f(4)=3,$ find $f^{-1}(1),$ $f^{-1}(0)$ and $f^{-1}(-1).$

  6. If If MATH

    1. Sketch the graph of $y=f(x).$

    2. Find $f(1)$

    3. Find MATH

  7. If MATH

    1. Sketch the graph of $y=f(x).$

    2. Find $f(2)$

    3. Find MATH