MATH 151. SUMMER 1999.

Review 2

1.

Using the definition of the derivative, find the derivative of

(a)

$f(x)=\sqrt{2x+1}$.

(b)

$f(x)=\sin 2x$.

2.

Find $\lim_{x\rightarrow 0} \frac{\cos 3x - \cos x}{\sin 5x}$.

3.

Calculate $y^\prime$:

(a)

$y=\frac{(2x+1)^3}{(3x+2)^2}$.

(b)

$y=\sqrt{2x^2+1}\cos^3 2x$.

(c)

$y=\tan(\cos \frac{x^2+1}{x^2})$.

(d)

$x\sin y+ y\sin x=2$.

4.

Find the equation of the tangent line to the curve $x+\tan xy =2$ at the point $(1,\frac{\pi}{4})$.

5.

Let the position of a particle at the time t is x(t)=t³-3t+1. Find the total distance traveled from t=0 to t=2.

6.

The angle of elevation of the sun increases at a rate of 0.1 rad/h. How fast is the shadow of 200-feet-tall building decreases when the length of the shadow is 400 ft?

7.

Without using calculator, find approximately $\sin \frac{401\pi}{100}$.

8.

The radius of a ball is 1 cm with a maximum error 0.1 cm. Estimate the maximum error in the calculated volume.