Calculus for a function of two inputs: the function and its domain, the surface it graphs, cross-sections and contour plots, and the partial derivatives that measure its rate of change in each direction, computed with the differentiation rules you already know.
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Lesson 03 · Level curves and contour plots
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01
Functions of two variables
12 questions
Learn what a function of two variables is and how to evaluate one at an ordered pair. Then find the domain as a region of the -plane, and decide whether its boundary curve belongs to the domain.
Covers
Evaluating a function of two variables
The domain as a region of the -plane
The boundary of the domain
02
Graphs and cross-sections of a function of two variables
8 questions
Learn how to picture a function of two variables as a surface in three dimensions, test whether a given point lies on that surface, and fix one input to cut the surface into a familiar one-variable curve.
Covers
The graph of as a surface
Cross-sections of a surface
03
Level curves and contour plots
8 questions
Learn how a horizontal plane cuts a surface into a level curve, and how to find that curve by solving . Then read a contour plot: which way is uphill, where the surface is steep or shallow, and where its peaks and basins lie.
Covers
Level curves
Reading a contour plot
04
What is a partial derivative?
10 questions
Learn what a partial derivative is: the rate at which a function of multiple variables changes as one input moves and the others are held fixed. Then read the signs, zeros and relative sizes of partial derivatives straight off a contour plot.
Covers
The partial derivative at a point
Reading a partial derivative from a graph
05
Computing partial derivatives
14 questions
Learn how to compute a partial derivative: evaluate one at a point by fixing the held input first, then produce it as a formula by treating the other variable as a constant. The same method extends to functions of three or more inputs, where every variable except the one named is held fixed.
Covers
Evaluating a partial derivative at a point
Computing a partial derivative
Partial derivatives of functions of more than two variables
06
Product and chain rules for partial derivatives
12 questions
Learn when a partial derivative needs the product rule and how to apply it. Then differentiate composite functions where the variable held constant enters through the inner derivative.
Covers
Differentiating a product
Differentiating a composite
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Second-order partial derivatives
In drafting
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6 lessons in this unit
74 prerequisite lessons across the unit, counting every step back to the start
66 other lessons in the learning path - after this unit, or alongside it
Builds on: Derivatives
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More in Calculus: Limits, Derivatives, Integrals