Calculus III Advanced (Course) (13.1) (Homework)

Section 13.1 Homework

From Calculus 10e by Larson and Edwards, p. 876. Exercises 16, 24, 27, 38, 45, 46, 47, 48, 52.

Exercise 13.1.16 Evaluating a Function

Find and simplify the function values.

$$ g(x,y) = \int_{x}^{y} \frac{1}{t} \: dt$$

(a) \((4,1)\) (b) \((6,3)\) (c) \((4, 3/2)\) (d) \((1/2, 7)\)
Solution
Integrate the function.[1]

$$ \int_{x}^{y} \frac{1}{t} \: dt = \ln|y| - \ln (x) $$

(a)	\(= \ln|1| - \ln (4) \)	\(= - 1.38629436112 \)
(b)	\(= \ln|3| - \ln (6) \)	\(= -0.405465108108 \)
(c)	\(= \ln|3/2| - \ln (4) \)	\(= -0.836988216786 \)
(d)	\(= \ln|1/2| - \ln (7) \)	\(= -1.94591014906 \)

Exercise 13.1.24 Finding the Domain and Range for a Function

Find the domain and range for the function.

$$ z = \frac{xy}{x-y} $$

Solution The function \(z\) is defined for all points \((x,y)\) such that \(x-y \ne 0 \). The domain is\( ( - \infty \leqslant xy \leqslant +\infty )\). The range is \((x-y > 0)\).

Exercise 13.1.27 Finding the Domain and Range for a Function

Find the domain and range for the function.

\(f(x,y) = \arccos (x+y) \)

Solution Domain for \(f(x,y) \) is \( -1 \leqslant x+y \leqslant 1 \). Range is \(0 \leqslant z \leqslant \pi \).

Exercise 13.1.38 Sketch a Surface

Sketch the surface given by the function.

$$ z = \frac{1}{2} \sqrt{x^{2}+y^{2}} $$

Solution The sketch is in Figure 1.[2]

Figure 1

Exercise 13.1.45 Contour Matching

Match the graph for the given surface with a contour map.

\( f(x,y)= e^{1-x^{2}-y^{2}} \)

Solution The graph most closely matches contour map c.

Exercise 13.1.46 Contour Matching

Match the graph for the given surface with a contour map.

\( f(x,y)= e^{1-x^{2}+y^{2}} \)

Solution The graph most closely matches contour map d.

Exercise 13.1.47 Contour Matching

Match the graph for the given surface with a contour map.

\( f(x,y)= \ln|y-x^{2}| \)

Solution The graph most closely matches contour map b.

Exercise 13.1.48 Contour Matching

Match the graph for the given surface with a contour map.

$$ f(x,y)= \cos \left( \frac{x^{2}+2y^{2}}{4} \right) $$

Solution The graph most closely matches contour map a.

Exercise 13.1.52 Sketch a Contour Map

Sketch a contour map for the surface using level curves for the given \(c\)-values.

\(f(x,y)=\sqrt{9-x^{2}-y^{2}}, \:\:\:\: c=0,1,2,3 \)

Solution Figure 6 shows the contours.[3]

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Parent Article: Calculus III Advanced (Course)