Mathematical Scratchpad

Practices

  1. For $f(x)=x^{2}+3x.$

    1. Use $h=0.001$ and $h=-0.001$ to find MATH respectively.

    2. Use the definition of $f$ MATH to prove that $f$ MATH

    3. Find the tangent line equation at the point $(1,4).$

  2. For $f(x)=-2x^{2}+3x.$

    1. Use $h=0.001$ and $h=-0.001$ to find MATH respectively.

    2. Use the definition of $f$ MATH to prove that $f$ MATH

    3. Find the tangent line equation at the point $(1,1).$

  3. Use the short cuts to find the derivatives for the following functions:

    1. MATH

    2. MATH

    3. MATH

  4. Find two functions which are continuous everwhere but are not differentiable at some point.

  5. Suppose $P(x)$ is a profit function (where $x$ denotes the number of units produced). Explain the followings:

    1. If MATH and $x_{1}<x_{2}$ then sketch $y=P(x)$ for this scenario. Should one increase or decrease the number of units produced? Explain.

    2. If MATH and $x_{1}<x_{2}$ then sketch $y=P(x)$ for this scenario. Should one increase or decrease the number of units produced?

  6. If MATH represent the cost function. Explain.

    1. Find the exact cost for the 34th unit. [hint: $C(34)-C(33)]$

    2. Find the marginal cost.

    3. Use the marginal cost to estimate the cost for the $34th$ unit. [hint: Use $C^{\prime }(33)].$

    4. Find the number of units produced which will either maximize or minimize the cost.

  7. If MATH represent the profit function.

    1. Find the marginal profit.

    2. What is profit (or loss) initially (when $x=0)$?

    3. Use the marginal profit to estimate the profit for the $32nd$ unit.

    4. Find the number of units produced which will maximize the profit.

    5. What is the maximum profit?