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Chapter 12: Problem 46

Find the first partial derivatives of the following functions. $$g(x, y, z)=2 x^{2} y-3 x z^{4}+10 y^{2} z^{2}$$

Short Answer

Expert verified
Question: Find the first partial derivatives of the function $$g(x, y, z) = 2 x^{2} y - 3 x z^{4} + 10 y^{2} z^{2}$$ with respect to x, y, and z. Answer: The first partial derivatives of the function g(x, y, z) are: $$\frac{\partial g}{\partial x} = 4xy - 3z^4,$$ $$\frac{\partial g}{\partial y} = 2x^2 + 20yz^2,$$ $$\frac{\partial g}{\partial z} = -12xz^3 + 20y^2z$$.

Step by step solution

01

Calculate 鈭俫/鈭倄

To find the partial derivative of the function g(x, y, z) with respect to x, we treat y and z as constants and differentiate g with respect to x: $$\frac{\partial g}{\partial x} = \frac{\partial}{\partial x}(2 x^{2} y - 3 x z^{4} + 10 y^{2} z^{2})$$ The derivative of each term is as follows: - The derivative of \(2x^2y\) with respect to x is \(4xy\) - The derivative of \(-3xz^4\) with respect to x is \(-3z^4\) - Since it does not contain x, the term \(10y^2z^2\) has no contribution. Therefore, $$\frac{\partial g}{\partial x} = 4xy - 3z^4$$.
02

Calculate 鈭俫/鈭倅

To find the partial derivative of the function g(x, y, z) with respect to y, we treat x and z as constants and differentiate g with respect to y: $$\frac{\partial g}{\partial y} = \frac{\partial}{\partial y}(2 x^{2} y - 3 x z^{4} + 10 y^{2} z^{2})$$ The derivative of each term is as follows: - The derivative of \(2x^2y\) with respect to y is \(2x^2\) - Since it does not contain y, the term \(-3xz^4\) has no contribution. - The derivative of \(10y^2z^2\) with respect to y is \(20yz^2\) Therefore, $$\frac{\partial g}{\partial y} = 2x^2 + 20yz^2$$.
03

Calculate 鈭俫/鈭倆

To find the partial derivative of the function g(x, y, z) with respect to z, we treat x and y as constants and differentiate g with respect to z: $$\frac{\partial g}{\partial z} = \frac{\partial}{\partial z}(2 x^{2} y - 3 x z^{4} + 10 y^{2} z^{2})$$ The derivative of each term is as follows: - Since it does not contain z, the term \(2x^2y\) has no contribution. - The derivative of \(-3xz^4\) with respect to z is \(-12xz^3\) - The derivative of \(10y^2z^2\) with respect to z is \(20y^2z\) Therefore, $$\frac{\partial g}{\partial z} = -12xz^3 + 20y^2z$$. Finally, the first partial derivatives of the function g(x, y, z) are: $$\frac{\partial g}{\partial x} = 4xy - 3z^4,$$ $$\frac{\partial g}{\partial y} = 2x^2 + 20yz^2,$$ $$\frac{\partial g}{\partial z} = -12xz^3 + 20y^2z$$.

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Most popular questions from this chapter

Suppose that in a large group of people, a fraction \(0 \leq r \leq 1\) of the people have flu. The probability that in \(n\) random encounters you will meet at least one person with flu is \(P=f(n, r)=1-(1-r)^{n} .\) Although \(n\) is a positive integer, regard it as a positive real number. a. Compute \(f_{r}\) and \(f_{n}.\) b. How sensitive is the probability \(P\) to the flu rate \(r ?\) Suppose you meet \(n=20\) people. Approximately how much does the probability \(P\) increase if the flu rate increases from \(r=0.1\) to \(r=0.11(\text { with } n \text { fixed }) ?\) c. Approximately how much does the probability \(P\) increase if the flu rate increases from \(r=0.9\) to \(r=0.91\) with \(n=20 ?\) d. Interpret the results of parts (b) and (c).

A function of one variable has the property that a local maximum (or minimum) occurring at the only critical point is also the absolute maximum (or minimum) (for example, \(f(x)=x^{2}\) ). Does the same result hold for a function of two variables? Show that the following functions have the property that they have a single local maximum (or minimum), occurring at the only critical point, but that the local maximum (or minimum) is not an absolute maximum (or minimum) on \(\mathbb{R}^{2}\). a. \(f(x, y)=3 x e^{y}-x^{3}-e^{3 y}\) b. \(f(x, y)=\left(2 y^{2}-y^{4}\right)\left(e^{x}+\frac{1}{1+x^{2}}\right)-\frac{1}{1+x^{2}}\) This property has the following interpretation. Suppose that a surface has a single local minimum that is not the absolute minimum. Then water can be poured into the basin around the local minimum and the surface never overflows, even though there are points on the surface below the local minimum. (Source: Mathematics Magazine, May 1985, and Calculus and Analytical Geometry, 2nd ed., Philip Gillett, 1984)

Use the Second Derivative Test to prove that if \((a, b)\) is a critical point of \(f\) at which \(f_{x}(a, b)=f_{y}(a, b)=0\) and \(f_{x x}(a, b)<0

The output \(Q\) of an economic system subject to two inputs, such as labor \(L\) and capital \(K,\) is often modeled by the Cobb-Douglas production function \(Q(L, K)=c L^{a} K^{b} .\) Suppose \(a=\frac{1}{3}, b=\frac{2}{3},\) and \(c=1\). a. Evaluate the partial derivatives \(Q_{L}\) and \(Q_{K}\). b. Suppose \(L=10\) is fixed and \(K\) increases from \(K=20\) to \(K=20.5 .\) Use linear approximation to estimate the change in \(Q\). c. Suppose \(K=20\) is fixed and \(L\) decreases from \(L=10\) to \(L=9.5 .\) Use linear approximation to estimate the change in \(\bar{Q}\). d. Graph the level curves of the production function in the first quadrant of the \(L K\) -plane for \(Q=1,2,\) and 3. e. Use the graph of part (d). If you move along the vertical line \(L=2\) in the positive \(K\) -direction, how does \(Q\) change? Is this consistent with \(Q_{K}\) computed in part (a)? f. Use the graph of part (d). If you move along the horizontal line \(K=2\) in the positive \(L\) -direction, how does \(Q\) change? Is this consistent with \(Q_{L}\) computed in part (a)?

Find an equation of the plane passing through (0,-2,4) that is orthogonal to the planes \(2 x+5 y-3 z=0\) and \(-x+5 y+2 z=8\)
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