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Calculus Post Process Tool \hspace{2.2in} {\it Raising Calculus to the Surface}
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\begin{enumerate}
\item Suppose the result of $\int_c^d \int_a^b q(x,y) \; dx \; dy$ is a real number, $k$. Explain what $k$ means and how it was measured. Sketch any images you have in mind in the space below.
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\item Suppose that a function $G$ is defined: $G = \int_1^t \int_{-4}^3 f(x) \; dx \; dy$
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\item $G$ is a function of which variable(s)? Justify your response.
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\item Now suppose that you increase the value of $t$. What would you need to know about the function $f$ to determine if $G$ increases or decreases?
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\item If $f(x) = xy - 3y^2$
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\item Determine the equation of the linear approximation $L(x,y)$ at the point $(x,y)=(2,3)$.
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\item What is the relationship between $f(x,y)$ and $L(x,y)$?
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\item At the point $(x,y) = (1.8, 3.1)$, the error of the approximation is $.05$. What does the $.05$ mean and why does it occur?
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Calculus Post Process Tool \hspace{2.2in} {\it Raising Calculus to the Surface}
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\item A function $f(x,y)$ is defined for all $x$ and $y$ greater than or equal to zero.
Describe all the ways you would use for finding the maximum or minimum values of the function. How do you know whether $f(x,y)$ has a maximum or minimum at a point?
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\item Suppose you have a function $f$ of two variables.
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\item What does it mean for $f$ to have a rate of change of $2.75$ for a value of $x=a$ and $y=b$? What information do you need to know to determine this value?
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\item If you were to explain to someone how to determine the rate of change of a two variable function, what would you say?
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\item Consider the phrase {\it the volume of a cylinder is a function of its circumference and height}.
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\item Please explain the meaning of this phrase.
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\item What values are in the domain of this function? What are possible units for these values?
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\item Create a symbolic representation of this function.
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\item Suppose $f(x,y)$ is a function. Describe the relationship between this function and $f(x+2,y-3)$.
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\item What is a function?
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\item What is a derivative?
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\item What is an integral?
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