Practices

1. For

1. Use and to find respectively.

2. Use the definition of to prove that

3. Find the tangent line equation at the point

2. For

1. Use and to find respectively.

2. Use the definition of to prove that

3. Find the tangent line equation at the point

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

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

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

1. If and then sketch for this scenario. Should one increase or decrease the number of units produced? Explain.

2. If and then sketch for this scenario. Should one increase or decrease the number of units produced?

6. If represent the cost function. Explain.

1. Find the exact cost for the 34th unit. [hint:

2. Find the marginal cost.

3. Use the marginal cost to estimate the cost for the unit. [hint: Use

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

7. If represent the profit function.

1. Find the marginal profit.

2. What is profit (or loss) initially (when ?

3. Use the marginal profit to estimate the profit for the unit.

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

5. What is the maximum profit?