Chapter 11: Problem 19
Describe the domain and range of the function. $$ f(x, y)=\arcsin (x+y) $$
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Find \(d^{2} w / d t^{2}\) using the appropriate Chain Rule. Evaluate \(d^{2} w / d t^{2}\) at the given value of \(t\) \(w=\frac{x^{2}}{y}, \quad x=t^{2}, \quad y=t+1, \quad t=1\)
True or False? In Exercises 59-62, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x, y)=\sqrt{1-x^{2}-y^{2}}\), then \(D_{\mathbf{u}} f(0,0)=0\) for any unit vector \(\mathbf{u}\).
Differentiate implicitly to find the first partial derivatives of \(z\) \(x+\sin (y+z)=0\)
The temperature at the point \((x, y)\) on a metal plate is modeled by \(T(x, y)=400 e^{-\left(x^{2}+y\right) / 2}, x \geq 0, y \geq 0\) (a) Use a computer algebra system to graph the temperature distribution function. (b) Find the directions of no change in heat on the plate from the point (3,5) . (c) Find the direction of greatest increase in heat from the point (3,5)
Use the function $$f(x, y)=3-\frac{x}{3}-\frac{y}{2}$$ Find \(\nabla f(x, y)\)
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