Functions of two variables and partial derivatives practice question 13
The surface is sliced by the vertical plane . The slice is the curve , drawn below with across and up. Which statement about the slice at the marked point, where , is correct?
Along the slice, is fixed at 1. Write as a function of alone. Which partial derivative is its slope?
. Differentiate in and put , then compare with at .
Theory UBC Math 100 §15.1Key results
Worked solution
Strategy
A vertical slice y = b holds y fixed, so the slice is the graph of the one-variable function f(x, b), and its slope at x = a is exactly f_x(a, b): the steepness of the surface in the x direction.
Working
The slice: put y = 1 and keep x.
The slope of the slice at x = 2: it falls 3 units of z per unit of x.
The same number: the slope of the slice y = 1 is f_x along it.
f_y would be the slope of the other slice through the point, x = 2, which this picture does not show.
Answer
The answer is The slice falls with slope −3 at x = 2, and that slope is f_x(2, 1)..
Check
f(2.1, 1) = 8 + 2.1 − 4.41 = 5.69, a drop of 0.31 from 6 over a step of 0.1 in x, close to a slope of −3.
Where this goes wrong
Reading the slice taken at y = 1 as information about y. Fixing y = 1 freezes y, so the curve shows only how f changes as x moves: its slope is f_x, and it says nothing about f_y.
The idea behind it
The functions of two variables and partial derivatives lesson, a figure to move, in a few steps
Leans on
Slope of a tangent line, from The definition of the derivative
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