engineering math2

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Applied Engineering Mathematics

Differentiation Rules

Proof?






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Differentiation Rulesrnek


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Derivation as a Rate of ChangeInstataneous Rate of Change

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Derivation as a Rate of Change


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Derivation as a Rate of Changernek








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Derivatives of Trigonometric Functions


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Derivatives of Trigonometric Functionsrnek




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The Chain Rule and Parametric Equations

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The Chain Rule and Parametric Equationsrnek

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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.








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Implicit Differentiation


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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.



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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.


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



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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.




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Applications of derivatives


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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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








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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.



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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.


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?





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Applied Engineering Mathematics Applied Engineering Mathematics



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