MATH 1150

Midterm 3 | Spring '13 | Dupre

Problem 1

Suppose that the function $f : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is given by $f(x, y) = x^2y - xy^2 = xy(x - y)$ We can easily see that $f(3, 2) = 6$ Therefore the point $(3, 2)$ is on the curve with equation $f(x, y) = 6$ which is a curve in the $(x, y)$-plane.

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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 $z = x^2y - xy^2 = z(t) = f(x(t), y(t)) = [x(t)]^2y(t) - x(t)[y(t)]^2, t \in \mathbb{R}$ Use the chain rule to express $\dot{z}$ in terms of $x, y, \dot{x}, \dot{y}$.

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Problem 3

Suppose that a beetle is crawling along the graph of the curve $z = f(x, y) = 6$ so that the rate of increase of $x$ for the beetle is $3$ units per second at the instant the beetle passes through the point $(3, 2)$. What is the rate of increase of $z$ for the beetle at the instant the beetle passes through the point $(3, 2)$?

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Problem 4

Suppose that a beetle is crawling along the graph of the curve $z = f(x, y) = 6$ so that the rate of increase of $x$ for the beetle is $3$ units per second at the instant the beetle passes through the point $(3, 2)$. What is the rate of increase of $y$ for the beetle at the instant the beetle passes through the point $(3, 2)$?

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Problem 5

What is $\frac{dy}{dx}$ at the point $(3, 2)$ on the curve $z = f(x, y) = 6$?

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Suppose that the function $f$ with domain $[0, 6]$ is defined by $y = f(x) = x(6 - x), 0 \le x \le 6$ Suppose that $R(x)$ is a rectangle with two sides on the coordinate axes and with a vertex at the origin $(0, 0)$ and with the vertex opposite the origin at the point $(x, y) = (x, f(x)), 0 \le x \le 6$. Let $A$ denote the area of the rectangle as a function of $x$, so $A(x)$ denotes the area of $R(x)$ for $0 \le x \le 6$. Then $A(x) = xy = xf(x) = x^2(6 - x) = 6x^2 - x^3, 0 \le x \le 6$

Problem 6

What is $A'(x)$ for $0 \lt x \lt 6$?

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Problem 7

What are the critical points of $A$ on the interval $0 \le x \le 6$?

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Problem 8

What is $A''(x)$ on the interval $0 < x < 6$?

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Problem 9

What are the inflection points of $A$ on the interval $0 < x < 6$?

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Problem 10

What is the value of $x$ for which the area $A(x)$ is maximum for $0 \le x \le 6$?

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Problem 11

Calculate the following integral. $\int_0^4 \sqrt{x} dx$

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Calculate the following derivative. $\frac{d}{dx}\int_{-2}^x [\sqrt{t^4 + t^2 + 2}] dt$

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