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NAME: ___________________________________________________________________ FINAL

1. (5 points) Find the area of the region enclosed by a parabola and a line

. To get full credit, you must draw a picture of the problem first, then find the upper and lower bounds before finding the area.

(2 ½ points each) Use appropriate logarithmic properties to make the following equation easy to differentiate. Find (and use trig identities to simplify to the simplest answer for full credit.

Use SUBSTITUTION method to find the answers of the given integrals. To get full credit, you need to show U, dU/dx, and solve for dx, and substitution, and the process of substitution.

(2 ½ points)

(inverse trig functions)

(5 points)

8. (5 points) Find the equation of a line that is tangent to a cure at ( 1 , 2 ). Leave the answer in the slope and y-intercept form as y = mx+b. No need to graph any picture at all.

9. (5 points) A 10 feet long ladder that was leaning against the wall as shown below. A guy was pushing the lower end of the ladder toward the wall and his action made the top end of the ladder moving up against the wall at the rate of 4 feet per second. How fast was the lower end of the ladder moving toward the wall when the lower end was 8 feet away from the wall? and at the moment what was the rate of the area of the right triangle that was formed by the ladder with the wall and the ground was decreasing? (NOTE for full credit: label all sides as A= the wall, B = the ground and C = the ladder)

10.
(5 points) You want to maximize the area of an inscribed rectangle under the line in the FIRST quadrant with the x-axis and the y-axis. Find the measurements of such rectangle so that you can have maximum area. Please draw the picture of the problem on x-y coordinate system to receive full credit.

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