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Applied Engineering Mathematics
Differentiation Rules
Proof?
Applied Engineering Mathematics
Differentiation Rules
Applied Engineering Mathematics
Differentiation Rules
Applied Engineering Mathematics
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Differentiation Rules
Applied Engineering Mathematics
Differentiation Rules
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Applied Engineering Mathematics
Differentiation Rulesrnek
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Applied Engineering Mathematics
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Differentiation Rules
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Applied Engineering Mathematics
Differentiation Rules
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Applied Engineering Mathematics
Differentiation Rules
Applied Engineering Mathematics
Differentiation Rules
Applied Engineering Mathematics
Derivation as a Rate of ChangeInstataneous Rate of Change
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Applied Engineering Mathematics
Derivation as a Rate of Change
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Applied Engineering Mathematics
Derivation as a Rate of Change
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Applied Engineering Mathematics
Derivation as a Rate of Changernek
Applied Engineering Mathematics
Derivation as a Rate of Change
Applied Engineering Mathematics
Derivation as a Rate of Change
Applied Engineering Mathematics
Derivation as a Rate of Change
Applied Engineering Mathematics
rnekDerivation as a Rate of Change
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Applied Engineering Mathematics
Derivatives of Trigonometric Functions
Applied Engineering Mathematics
Derivatives of Trigonometric Functionsrnekler
Applied Engineering Mathematics
Derivatives of Trigonometric Functions
Applied Engineering Mathematics
Derivatives of Trigonometric Functionsrnek
Applied Engineering Mathematics
Derivatives of Trigonometric Functions
Applied Engineering Mathematics
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Derivatives of Trigonometric Functions
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Applied Engineering Mathematics
The Chain Rule and Parametric Equations
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Applied Engineering Mathematics
The Chain Rule and Parametric Equationsrnek
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Applied Engineering Mathematics
The Chain Rule and Parametric Equations
Applied Engineering Mathematics
The Chain Rule and Parametric Equationsrnek
Applied Engineering Mathematics
The Chain Rule and Parametric Equations
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Applied Engineering Mathematics
The Chain Rule and Parametric Equationsrnek
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Applied Engineering Mathematics
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The Chain Rule and Parametric Equations
Applied Engineering Mathematics
The Chain Rule and Parametric Equations
Parametric EquationsInstead of describing a curve by expressing the y-coordinate of a point P(x, y) on the curve as a
function of x, it is sometimes more convenient to describe the curve by expressing both
coordinates as functions of a third variable t.
Applied Engineering Mathematics
The Chain Rule and Parametric Equationsrnek
Applied Engineering Mathematics
The Chain Rule and Parametric Equationsrnek
Applied Engineering Mathematics
The Chain Rule and Parametric Equations
Applied Engineering Mathematics
rnekThe Chain Rule and Parametric Equations
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Applied Engineering Mathematics
The Chain Rule and Parametric Equationsrnek
Applied Engineering Mathematics
The Chain Rule and Parametric Equations
Applied Engineering Mathematics
Implicit Differentiation
Applied Engineering Mathematics
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Implicit Differentiation
Applied Engineering Mathematics
Implicit Differentiation
Applied Engineering Mathematics
Implicit Differentiation
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Applied Engineering Mathematics
Implicit Differentiation
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Applied Engineering Mathematics
Implicit Differentiation
Applied Engineering Mathematics
Implicit Differentiation
Using implicit differentiation in was much simpler than calculating dy/dx directly from any of
the above formulas. Finding slopes on curves defined by higher-degree equations usuallyrequires implicit differentiation.
Applied Engineering Mathematics
Implicit Differentiation
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Applied Engineering Mathematics
Implicit Differentiation
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Applied Engineering Mathematics
Implicit Differentiation
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Applied Engineering Mathematics
Implicit Differentiation
Applied Engineering Mathematics
Implicit Differentiation
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Applied Engineering Mathematics
Implicit Differentiation
Applied Engineering Mathematics
Linearization and DifferentialsWe introduce new variables dx / dy, called differentials, and define them in a waythatmakes Leibnizs notation for the derivative a true ratio. We use dy to estimate error inmeasurement and sensitivity of a function to change.
Applied Engineering Mathematics
Linearization and Differentials
In general, the tangent to y=f(x) at a point where is differentiable, passes through thepoint (a, (a)), so its point-slope equation
Applied Engineering Mathematics
Linearization and Differentials
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Applied Engineering Mathematics
Linearization and Differentials
Applied Engineering Mathematics
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Linearization and Differentials
Applied Engineering Mathematics
Linearization and Differentials
Applied Engineering Mathematics
Linearization and Differentials
Applied Engineering Mathematics
rnekThe radius r of a circle increases from to 10.1 m. Use dA to estimate the increase in thecircles areaA. Estimate the area of the enlarged circle and compare your estimate tothe true area.
Linearization and Differentials
Applied Engineering Mathematics
Linearization and Differentials
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Applied Engineering Mathematics
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Linearization and Differentials
Applied Engineering Mathematics
Applications of derivatives
Applied Engineering Mathematics
A drilling rig 12 mi. offshore is to be connected by pipe to a refinery onshore, 20miles straight down the coast from the rig. If underwater pipe costs 500,000 permile and land based pipe costs $300,000 per mile, what combination of the twowill give the least expensive connection?
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Applications of derivatives
Underwater pipe is more expensive, so weuse as little as we can. We run straight toshore (12 mi) and use land pipe for 20 mito the refinery.Dollar cost = 12*500,000 + 20*300,000
= = 12,000,000
Applied Engineering Mathematics
Applications of derivatives
Applied Engineering Mathematics
Applications of derivatives
Applied Engineering Mathematics
Applications of derivatives
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Applied Engineering Mathematics
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Applications of derivatives
is at x=1 zero at and undefinedat x=0. There are no endpointsin the domain, so the criticalPoints and are the only placeswhere might have an extremevalue.
Applied Engineering Mathematics
Applications of derivatives
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Applied Engineering Mathematics
Applications of derivatives
Applied Engineering Mathematics
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Applications of derivatives
Applied Engineering Mathematics
Applications of derivatives
Applied Engineering Mathematics
Applications of derivatives
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Applied Engineering Mathematics
Applications of derivatives
Applied Engineering Mathematics
Optimization ProblemsTo optimize something means to maximize or minimize some aspect of it. What are thedimensions of a rectangle with fixed perimeter having maximum area? What is the leastexpensive shape for a cylindrical can? What is the size of the most profitable productionrun? The differential calculus is a powerful tool for solving problems that call formaximizingor minimizing a function. In this section we solve a variety of optimizationproblems from business, mathematics, physics, and economics.
Applications of derivatives
rnekAn open-top box is to be made by cuttingsmall congruent squares from the cornersof a 12-in.-by-12-in. sheet of tin andbending up the sides. Howlarge should thesquares cut from the corners be to makethe boxhold as much as possible?
Applied Engineering Mathematics
Applications of derivatives
Applied Engineering Mathematics
Applications of derivatives
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Applied Engineering Mathematics
Applications of derivatives
Applied Engineering Mathematics
Applications of derivatives
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Applied Engineering Mathematics
Applications of derivatives
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Applied Engineering Mathematics
Applications of derivatives
Applied Engineering Mathematics Applied Engineering Mathematics
Applied Engineering Mathematics Applied Engineering Mathematics
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Applied Engineering Mathematics Applied Engineering Mathematics
Applied Engineering Mathematics Applied Engineering Mathematics
Applied Engineering Mathematics Applied Engineering Mathematics