MATH 151. SUMMER 1999.

Review 3

1.

Find the absolute and local maximum and minimum values for

(a)

y=x² (x+1)³ (x+2).

(b)

$y=\sin^2 x - x$.

2.

Find limits

(a)

$\lim_{x\rightarrow \infty} \frac{2x^2+2x+7}{3x^2+7x+2}$.

(b)

$\lim_{x\rightarrow \infty} \left( \sqrt{x^2+2x+3}-\sqrt{x^2+4} \right)$.

3.

Show that equation $2x=\cos x$ has exactly one root.

4.

Find asymptotes for $y=\sqrt{4x^2+x+3}+\frac{1}{x}$.

5.

For a given function, determine intercepts, symmetry, intervals of increase and decrease, local maximum and minimum values, concavity, asymptotes and sketch its graph:

(a)

$y=\frac{x^2}{x^2-9}$.

(b)

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

(c)

$y=\frac{1}{\sin x}$.

6.

Prove that $\sqrt{1+x}>1+\frac{1}{2}x-\frac{1}{8}x^2$ for x>0.

7.

Determine the height and the radius of the cylindrical pan (without a lid) of volume $1\;\hbox{ft}.^3$ which has the smallest surface area.