ALL PROBLEMS EQUALLY WEIGHTED

Problem 1. Using the definition of derivatives shown to the right, derive the derivative of g(x).

since:

Let:

we can easily show:

Let's just work on the first term where:

Using the generalized binomial expansion where for any value of n, whether positive, negative, integer or non-integer, the value of the nth power of a binomial is given by::

downloaded 3-2010 from: http://hyperphysics.phy-astr.gsu.edu/hbase/alg3.html (Georgia State University)

where:

Note: all terms will always cancel out except the 2nd term.

Rationalizing the denominator:

thus:

Problem 2. For h(x), find the equations of the tangent line(s) that pass through the point (-1,-10).

derive line equation by using point-slope form:

setting this equal to the parabola to find intersection;

find the roots of this quadratic yields:

placing the x -values of the intersections yields the tangent line equations:

Problem 3. Find:

given

Problem 4. Find:

given

Problem 5. Find:

given

Problem 6. Find:

given

Problem 7. Find a horizontal tangent of:

Provide a sketch of the function with these tangent lines.

(radians)

Problem 8. Evaluate the derivative of the function R(x) at the given point (3 decimal place accuracy):

(deg)

Problem 8. Evaluate the derivative of the function R(x) at the given point (3 decimal place accuracy):

Problem 9. Implicitly differentiate the equation to the right (with respect to x). Determine the slope of the line tangent to the equation at point A. Place the axes and sloped field(s) for this derivative on the graph below.

Graphing domain:

Explicitly differentiated:

Implicitly differentiating:

Slope at A

tangent line equation:

Sloped Fields

TI Screen Shot

Problem 10. An airplane is flying at an altitude of 5 miles and passes directly over a radar antenna. When the airplane is 10 miles away (s = 10 mi), the radar detects that the distance s is changing at a rate of 240 miles per hour. What is the speed of the plane?

miles/hour

11. The displacement position of a bicycle rider on HWY 50 (starting in Penrose) is shown below.

Trace the velocity curve in pencil on the Displacement Graph.

Where is the rider's position at 5.0 hours?

BONUS (5%) - What is the probability Mr. Heinen will retire this year?

A 0%

B 20%

C 60%

D 100%