Exam 14: Differentiating Functions of Several Variables

If a function z = g(x, y)has g(1, 2)= -5, g_x(1, 2)= 4 and g_y(1, 2)= 3, find the equation of the plane tangent to the surface z = g(x, y)at the point where x = 1 and y = 2.

Let $f ( x , y ) = x ^ { 2 } + 5 y ^ { 2 } - 5 x$ What is the direction of maximum rate of change of f at (1, 1)?

Find the quadratic approximation to the function f(x, y)= cos x cos y valid near the origin.

What is the angle between the unit vector $\vec { u }$ and $\nabla f ( a , b )$ if $f _ { a } ( a , b ) = - 0.5 \| \nabla f ( a , b ) \|$ ?

For the function g(x, y)with contour diagram below: Evaluate g(-1, 1).

Find the directional derivative of $f ( x , y ) = x ^ { 3 } y + 5 x y ^ { 2 }$ at the point (1, 1)in the direction of $3 \vec { i } + 5 \vec { j }$ .

If f and g have the same partial derivatives at every point in their domains, then $f = g$ .

Find the angle between the vector $\vec { c } = 2 \vec { i } + 3 \vec { j } - \vec { k }$ and the positive z-axis.

If $g ( x , y ) = x ^ { 2 } \cos ( x y )$ , then find $\frac { \partial g } { \partial y } \left( 3 , \frac { \pi } { 3 } \right)$ .

Let $f ( x , y , z ) = x ^ { 3 } + 6 x z + 4 y z + z ^ { 2 }$ .Find $\nabla f ( 1 , - 2,6 )$ .

Suppose that the maximum rate of change of f at $( 1 - 1 )$ is 20, and it occurs in the direction of $5 \vec { i } - 5 \vec { j }$ .Find $\nabla f ( 1 , - 1 )$ , $f _ { x } ( 1 , - 1 )$ , and $f _ { y } ( 1 , - 1 )$ .

Suppose that the maximum rate of change of f at $( 1 - 1 )$ is 10, and it occurs in the direction of $2 \vec { i } - 2 \vec { j }$ .What is the minimum rate of change of f at $( 1 - 1 )$ , and in which direction does it occur?

If $P is invested in a bank account earning r% interest a year, compounded continuously, the balance, $B, at the end of t years is given by $B = f ( P , r , t ) = P e ^ { r t / 100 }$ (a)What are the units of $\partial B / \partial t$ ? (b)What is the practical interpretation (in terms of money)of $\partial B / \partial t$ ?

Estimate $z _ { y } ( 3,2 )$ numerically if $z = ( x + y ) ^ { ( x + y ) } , x , y > 0$

The cross-sections of f when x is fixed at x = 1 and when y is fixed at y = 2 are given below. Determine, if possible, the sign of $f _ { y } ( 1,3 )$

The consumption of beef, C (in pounds per week per household)is given by the function C = f(I, p), where I is the household income in thousands of dollars per year, and p is the price of beef in dollars per pound. Do you expect $\partial f / \partial p$ to be positive or negative?

Use the differential of $f ( x , y ) = x ^ { 2 } + 2 x \cos ^ { 2 } y$ to find a linear approximation of f at the point (1, $\pi$ /4).

A gradient is a specific example of a directional derivative.

A ball thrown from ground level with initial speed v and at an angle $\alpha$ with the horizontal hits the ground at a distance $s = \frac { v ^ { 2 } \sin ( 2 \alpha ) } { g }$ away from the starting point (where g is the acceleration due to gravity).Calculate the differential ds.

Find a unit vector perpendicular to both $6 \vec { i } - \vec { j } + \vec { k }$ and $\overrightarrow { 8 i } + \vec { k }$ .