Exam 2

Show all work in a neat and organized fashion. Clearly indicate your answers. 100 points possible.

1. (10 pts.) Find the derivative of each function. (You do not have to simplify your answer.)

(a) f(x)=(3x-8)⁵(7x²-3x+1)¹⁴

(b) g(x)=(1+sin²x)⁷

2. (10 pts.) Find dy/dx by implicit differentiation.

y2=x2+sin(xy)

3. (10 pts.) A particle moves according to a law of motion

s=f(t)=t3-3t2+3t,       (t ³ 0)

where t is measured in seconds and s in meters.

(a) Find the acceleration at time t, and after 2 seconds.

(b) Graph the position, velocity, and acceleration functions for 0 £ t £ 3.

(c) When is the particle speeding up? When is it slowing down?

4. (10 pts.) A plane flying horizontally at an altitude of 1.5 miles and at a speed of 400 miles per hour passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 6 miles away from the station.

5. (10 pts.) Use differentials, showing all work, to estimate the following.

1 50.1

6. (10 pts.) Let f(x)=2x³-[31/2] x² +5x. Use calculus methods, showing all work, to do the following.

(a) Find the intervals of increase or decrease.

(b) Find the local maximum and minimum values.

f(x)=2x3- 31 2 x2 +5x

7. (10 pts.) Let f(x)=[1/12]x⁴-[5/2] x². Use calculus methods, showing all work, to do the following.

Find the intervals of concavity and the inflection points.

f(x)= 1 12 x4- 5 2 x2

8. (10 pts.) Find the dimensions of the rectangle of largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola y=16-x².

9. (5 pts.) Use Newton's method with the specified initial approximation x₁ to find x₂, the second approximation to the root of the given equation.

2x3+x2-x+1=0,       x1=-1.2

Recall that

x2=xn- f(x1) f¢(x1)

and

xn+1=xn- f(xn) f¢(xn) .

10. (5 pts.) Find f(x), given the following.

f¢(x)=x3+5sinx

11. (5 pts.) By reading values from the given graph of f, use five rectangles to find a lower estimate for the area under the given graph of f from x=0 to x=10. Sketch the rectangles that you use.

Figure

12. (5 pts.) The graph of f is shown. Evaluate the following integral by interpreting it in terms of areas.

ó õ 7 2  f(x) dx

Figure