## Practice Problems

Work through the odd-numbered problems 1-47. Once you have completed the problem set, check your answers.

### Problems

1. Let and . Find the values of , and at .

3. Estimate the values of and in Fig. 5 and determine

In problems 5-9, find the derivative of each function.

11. A weight attached to a spring is at a height of feet above the floor seconds after it is released.

(b) At what height is the weight when it is released?

(c) How high does the weight ever get above the floor and how close to the floor does it ever get?

(d) Determine the height, velocity and acceleration at time t. (Be sure to include the correct units.)

(e) Why is this an unrealistic model of the motion of a weight on a real spring?

13. The kinetic energy of an object of mass and velocity is .

(a) Find the kinetic energy of an object with mass and height feet at and seconds.

(b) Find the kinetic energy of an object with mass and height feet at and seconds.

In problems 15-19, find the derivatives .

In problems 21-25, find the equation of the line tangent to the graph of the function at the given point.

27. Find the equation of the line tangent to at the point . Where will this tangent line intersect the -axis? Where will the tangent line to at the point intersect the -axis?

In problems 29-33, calculate and

35. What will the derivative of a quadratic polynomial be? The derivative? The derivative?

37. What can you say about the and derivatives of a polynomial of degree ?

In problems 39-41, you are given . Find a function with the given derivative.

43. The function in Fig. 6 is continuous at 0 since . Is differentiable
at (Use the definition of and consider )

The number appears in a variety of unusual situations.

Problems 45 – 47 illustrate a few of them.

45. Use your calculator to examine the values of when is relatively large, for example, , and 10000 . Try some other large values for . If
is large, the value of is close to what number?

47. (a) Calculate the value of the sums , , and

(b) What value do the sums in part (a) seem to be approaching? Calculate and .

( product of all positive integers from 1 to n. For example, )

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-3.4-More-Differentiation-Problems.pdf

This work is licensed under a Creative Commons Attribution 3.0 License.