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MA35 Labs

1 » Single Variable Calculus

2 Two Variable Calculus

3 Three Variable Calculus

Polar Coordinate Labs

Two Variable Functions

Contents

1.1 Functions of One Variable

1.2 Differentiation

1.2.1 Derivatives

1.2.2 Critical Points

1.2.3 Tangent Lines and Normal Vectors

1.2.4 Chain Rule

1.2.5 Differentiability

1.3 More Derivatives

1.4 Integration

1.5 Fundamental Theorem of Calculus

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Critical Points (Page: 1 | 2 )

Text

A critical point of a parametrized curve (x(t), y(t)) in the plane is a value t₀ of t such that the tangent line is vertical or horizontal.

t₀ is a critical point if and only if x(t) is defined and x'(t) = 0 or y(t) is defined and y'(t) = 0 (or both).

Demos

Critical Points

In this demo, increasing x is indicated by red, decreasing x by absence of red, increasing y by blue, and decreasing y by absence of blue. Consequently, when x and y are both increasing, the red and blue will overlap in a purple section of the curve. The curve will be white where both x and y are decreasing. You can use these colors to find critical points. Any point at which the curve changes from red to white or purple (or vice versa) is a critical point for x(t) and any point at which the curve changes from blue to white or purple (or vice versa) is a critical point for y(t).