009A Sample Final 1
This is a sample, and is meant to represent the material usually covered in Math 9A for the final. An actual test may or may not be similar. Click on theto go to a solution.
In each part, compute the limit. If the limit is infinite, be sure to specify positive or negative infinity.
Consider the following piecewise defined function:
- a) Show that is continuous at .
- b) Using the limit definition of the derivative, and computing the limits from both sides, show that is differentiable at .
Find the derivatives of the following functions.
compute and find the equation for the tangent line at . You may leave your answers in point-slope form.
A kite 30 (meters) above the ground moves horizontally at a speed of 6 (m/s). At what rate is the length of the string increasing when 50 (meters) of the string has been let out?
Consider the following function:
- a) Use the Intermediate Value Theorem to show that has at least one zero.
- b) Use the Mean Value Theorem to show that has at most one zero.
A curve is defined implicitly by the equation
- a) Using implicit differentiation, compute .
- b) Find an equation of the tangent line to the curve at the point .
- a) Find the differential of at .
- b) Use differentials to find an approximate value for .
- Given the function ,
- a) Find the intervals in which the function increases or decreases.
- b) Find the local maximum and local minimum values.
- c) Find the intervals in which the function concaves upward or concaves downward.
- d) Find the inflection point(s).
- e) Use the above information (a) to (d) to sketch the graph of .
Consider the following continuous function:
defined on the closed, bounded interval .
- a) Find all the critical points for .
- b) Determine the absolute maximum and absolute minimum values for on the interval .
Contributions to this page were made by Kayla Murray