#### MATH 1150

##### Final | Spring '13| Dupre
1. Suppose that the function $f : R × R → R$ is given by

PROBLEM 2. Suppose that x and y are both functions of time, t, that $x = x(t)$, and $y = y(t)$, and the function $z = z(t)$ of time is given by $z = f(x, y)$, so that

Use the chain rule to express in terms of

PROBLEM 3. Suppose that a beetle is crawling along the graph of the curve

$z = f(x, y) = 8$

so that the rate of increase of x for the beetle is 6 units per second at the instant the beetle passes through the point (4, 2). What is the rate of increase of $z$ for the beetle at the instant the beetle passes through the point (4, 2)?

PROBLEM 4. Suppose that a beetle is crawling along the graph of the curve

$z = f(x, y) = 8$

so that the rate of increase of x for the beetle is 6 units per second at the instant the beetle passes through the point (4, 2). What is the rate of increase of $y$ for the beetle at the instant the beetle passes through the point (4, 2)?

PROBLEM 5. What is at the point (4, 2) on the curve $z = f(x, y) = 8$?

PROBLEM 6. What is

PROBLEM 7. What are the critical points of $g$ on the interval

PROBLEM 8. What is on the interval $−1 < x < 12$?

PROBLEM 9. What are the inflection points of &g& on the interval &−1 < x < 12&?

PROBLEM 10. What is the value of $x$ for which $g(x)$ is maximum for $−1 ≤ x ≤ 12$?

Calclate the following integrals and derivatives.

PROBLEM 11.

PROBLEM 12.