Second Midterm Solutions of MATH 113-A

1) Find all indicated derivatives of the following functions (35 points)

A) (7P)

B)

C) (7 P)

D) (7 Points)

E) (7 Points)

2. Find the Equation of the tangent line for function at x=0 (15 Points)

3) for the function show each of the following steps

A) The intervals on which f(x) is increasing and decreasing

-3 0

-

+

+

Therefore f(x) is decreasing when x<-3 and increasing when x>-3 (5 points)

B) The intervals on which f(x) is concave up and concave down

-2 0

+

-

+

Therefore f(x) is concave up in x<-2 and x>0 and concave down in -2<x<0 (5points)

C) All critical points are (-3) and (0)

D) Inflection Points are (0) and (2)

E)

-3 -2 0

	-	+	+	+
	+	+	-	+
f	CU	CU	CD	CU

4) A 13 meter ladder is leaning against a house when its base starts to slide away. By the time the base is 12 meters from the house, the base is moving at the rate of 5m/sec (25 points)

A) How fast the top of the ladder sliding down the wall then?

(dy/dx=?)

The length of the ladder is constant all the time but since the top and the bottom move then we can set them as y and x respectively, therefore

From the Pythagoras theorem

B) At what rate is the area of the triangle formed by the ladder, wall, and ground changing than?

The area is

C) At what rate is the angle between the ladder and the ground changing then?