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(MTH 250)
Lecture 24
Calculus
Previous Lecture’s Summary
•Multivariable functions
•Limits along smooth curves
•Limits of multivariable functions
•Continuity of multivariable functions
•Partial differentiation
Today’s Lecture
•Recalls
•Differentiability of multivariable functions
•Differentials & linear approximations
•Chain rules for partial differentiation
•Implicit differentiation
•Extrema of multivariable functions
Definition: In general, a function of variables, is a rule that assigns a unique real number to each point in some set in the . The set is called its domain and the set is called its range. Definition: If is a function of two variables with domain then the graph of is the set of all points in such that and is in .
Recalls
Defintion: If is a smooth parametric curve in 2D or 3D that is represented by
Then,
Definitions: Let and The set of all point that are enclosed by the circle centered at and radius , but do not lie on the circle, is called the open disk of radius and center The set of points that lie on the circle together with those enclosed by the circle is called the closed disk.
Recalls
Definitions: Analogously, if is a sphere centered at and has positive radius , then the set of points that enclosed by the sphere, but do not lie on the sphere, is called the open ball. The set of points that lie on the sphere together with those enclosed by the sphere is called the closed ball.
If is a set in , then called an interior point of if there is some open disk centered at that contains only points of D, and is called a boundary point if every open disk centered at contains both points in and points not in
A set is called closed if it contains all of its boundary points and open if it contains none of its boundary points. and are both closed and open.
Recalls
Definition (formal):
Recalls
Theorem:
Def:
Recalls
Recalls
Theorem: Let be a function of two variables. If and are continuous on some open disk, then on the disk
Recalls
Definition: Let be a function of two variables and let be a point on the graph of . Suppose and are the intersections of the surface with the plane and Let and be the tangent lines to the curvesand at point Then, if itexists, the plane containing and iscalled the tangent plane to at point
Tangent plane
• Recall that the equation of the plane in point-normal form passing through point is of the form
.• Assuming the plane is not vertical, we have .• The equation of the plane as • The tangent line to atisobtained by taking i.e.
• Since is the slope of the tangent line to at .• Therefore, Similarly, • Thus the equation of the tangent plane is
Tangent plane
Tangent planeExample:
Sol.
Definition:
Definition:
Differentiability of multivariable functions
Example: Prove that is differentiable at
Solution:
Differentiability of multivariable functions
Remarks: Define for
• Define • By definition of differentiability we have• Immediately by definition of we have
• In other words, if is differentiable at then may be expressed as shown above, where as and further when
Differentiability of multivariable functions
Differentiability of multivariable functionsTheorem: If a function is differentiable at a point, then it is continuous at that point.
Differentiability of multivariable functions
Theorem: If all first-order partial derivatives of exist and are continuous at a point, then is differentiable at that point.
Example: Consider the function Since
are defined and continuous everywhere.We conclude that is differentiable everywhere.
Definition (Total differential):
Differentials & linear approximations
Differentials & linear approximations
Example: Approximate the change in fromits value at to its value at Compare the magnitude of the error in this approximation with the distance between the points
Solution:
Differentials & linear approximations
Cont:
Differentials & linear approximations
Differentials & linear approximations
Differentials & linear approximations
Differentials & linear approximationsLocal linear approximation: Since the tangent plane to the surface at is very close to the surfact at least when it is near we may use the function defining the tangent plane as a linear approximation to Recall that the equation of the tangent plane to the graph of at is
Differentials & linear approximationsLocal linear approximation
Differentials & linear approximationsSolution:
Chain rules for partial differentiationDefinition (chain rules):
Chain rules for partial differentiation
Chain rules for partial differentiationDefinition (chain rules) for partial derivatives:
Chain rules for partial differentiation
Chain rules for partial differentiation
Chain rules for partial differentiationRemarks: Let is differentiable function of Then
• .
• When then
Theorem:
Chain rules for partial differentiation
Extrema of multivariable functions Definitions:
Extrema of multivariable functions Theorem (Extream value): If is continuous on a closed and bounded set , then has both an absolute maximum and an absolute minimum on
Definition: A point in the domain of a function is called a critical point of the function if if one or both partial derivatives do not exist at
Theorem: If has a relative extremum at a point if it is a critical point and the first order partial derivatives of exist at
Extrema of multivariable functions Saddle point: If has neither a relative maximums nor a relative minimum at a critical point , then it is called a saddle point of
In general, the surface has a saddle point at if there are two distinct vertical planes through this point such that the trace of the surface in on of the planes has a relative maximum and the trace in the other has a relative minimum at
Extrema of multivariable functions Theorem:
Extrema of multivariable functions
Extrema of multivariable functions
Extrema of multivariable functions
Lecture Summary
•Recalls
•Differentiability of multivariable functions
•Differentials & linear approximations
•Chain rules for partial differentiation
•Implicit differentiation
•Extrema of multivariable functions
