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#### Fourier Series Before looking at the example, you need to know the formula of Fourier Series as shown. If you know the additional information shown, you could reduce your work.  (1) Odd functions have Fourier Series with only sine terms, which means you only find the coefficients, b’s in the formula.  (2) Even functions have Fourier Series...

#### Derivatives of Logarithmic Functions Before starting examples, you need to know the derivative formulas as shown. In many cases, we need to make use of the properties of logarithm as well. Please remember that if you see “ln” symbol, it is called natural log and it has the base “e”. If you see “log”, it has the base 10.

#### 2nd order Nonhomogeneous Differential Equation Before looking at the example, you need to know the solution formula for second-order differential equations ay”+ by’ +cy = f(x) as shown.  Notice that there are two parts, y-sub C and y-sub P in the complete solution. One part, y-sub C is solving a homogeneous differential equation.  Y-sub C is often called a complementary...

#### Integrating Rational Fractions by Partial Fractions If you take a look at the integrand of the question, it seems a relatively complicated fraction.  If we can split it into simpler fractions, then we may be able to integrate them easily. Making use of partial fractions to get the simpler fractions. First, we need to have factored form of the denominator to...

#### Integration By Parts In this example, there is a product of x^2 and cosx dx in the integrand. Does simple u-substitution method work for this example? In order to use the simple u-substitution method, the relationship between two functions must be the original and its derivative each other except for constants. Unfortunately, x^2 cannot be the derivative of...

#### Optimization Draw a picture of the scenario, if you can. Step(1) Formulate the objective function. Step(2) Reduce the objective function to One variable. Step(3) Take the derivative of the function and find the critical value(s). Step(4) Find the local max or min by either First derivative (for rational         fractions ) or Second derivative test...
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