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Integral Calculus
Gottfried Wilhelm Leibnitz (1646-1716) and Sir Isaac Newton (1643-1727) independently discovered calculus in the mid-17th…
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Lagrange Multipliers
In the previous section we optimized (i.e. found the absolute extrema) a function on a region…
Posted inMathematics for Marine Engineering
Finding Maxima and Minima using Derivatives
Where is a function at a high or low point? Calculus can help! A maximum…
Posted inMathematics for Marine Engineering
Examples on Chain Rule (Differentiation Rules)
Examples on Chain Rule Example 10.8 Find F′(x) if F (x) = √[x2 +1] Solution Take u = g(x) = x2 + 1 and f (u)…
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Euler’s Homogeneous Function Theorem
Let be a homogeneous function of order so that (1) Then define and . Then (2)(3)(4) Let , then (5) This can…
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Homogeneous Functions
To be Homogeneous a function must pass this test: f(zx,zy) = znf(x,y) In other words Homogeneous is when…
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Higher-Order Partial Derivatives
Fred has mastered differentiating when a function has only one variable. Somehow, he missed the…
Posted inMathematics for Marine Engineering
Partial Derivatives
Now that we have the brief discussion on limits out of the way we can…
Posted inMathematics for Marine Engineering
Limits and Continuity
The concept of the Limits and Continuity is one of the most crucial things to understand…