CALCULUS

• NOTE: Calculus is very useful when looking at real systems. Many students are turned off by the topic because they "don’t get it". But, the secret to calculus is to remember that there is no single "truth" - it is more a loose collection of tricks and techniques. Each one has to be learned separately, and when needed you must remember it, or know where to look.

35.6.1 Single Variable Functions

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35.6.1.1 - Differentiation

• The basic principles of differentiation are,

 

• Differentiation rules specific to basic trigonometry and logarithm functions

 

• L’Hospital’s rule can be used when evaluating limits that go to infinity.

 

• Some techniques used for finding derivatives are,

 

35.6.1.2 - Integration

• Some basic properties of integrals include,

 

• Some of the trigonometric integrals are,

 

• Some other integrals of use that are basically functions of x are,

 

 

• Integrals using the natural logarithm base ‘e’,

 

35.6.2 Vector Calculus

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• When dealing with large and/or time varying objects or phenomenon we must be able to describe the state at locations, and as a whole. To do this vectors are a very useful tool.

• Consider a basic function and how it may be represented with partial derivatives.

 

• Gauss’s or Green’s or divergence theorem is given below. Both sides give the flux across a surface, or out of a volume. This is very useful for dealing with magnetic fields.

 

• Stoke’s theorem is given below. Both sides give the flux across a surface, or out of a volume. This is very useful for dealing with magnetic fields.

 

35.6.3 Differential Equations

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• Solving differential equations is not very challenging, but there are a number of forms that need to be remembered.

• Another complication that often occurs is that the solution of the equations may vary depending upon boundary or initial conditions. An example of this is a mass spring combination. If they are initially at rest then they will stay at rest, but if there is some disturbance, then they will oscillate indefinitely.

• We can judge the order of these equations by the highest order derivative in the equation.

• Note: These equations are typically shown with derivatives only, when integrals occur they are typically eliminated by taking derivatives of the entire equation.

• Some of the terms used when describing differential equations are,

ordinary differential equations - if all the derivatives are of a single variable. In the example below ’x’ is the variable with derivatives.

 

first-order differential equations - have only first-order derivatives,

 

second-order differential equations - have at least on second derivative,

 

higher order differential equations - have at least one derivative that is higher than second-order.

partial differential equations - these equations have partial derivatives

• Note: when solving these equations it is common to hit blocks. In these cases backtrack and try another approach.

• linearity of a differential equation is determined by looking at the dependant variables in the equation. The equation is linear if they appear with an exponent other than 1.

 

35.6.3.1 - First-order Differential Equations

• These systems tend to have a relaxed or passive nature in real applications.

• Examples of these equations are given below,

 

• Typical methods for solving these equations include,

guessing then testing

separation

homogeneous

35.6.3.1.1 - Guessing

• In this technique we guess at a function that will satisfy the equation, and test it to see if it works.

 

• The previous example showed a general solution (i.e., the value of ’C’ was not found). We can also find a particular solution.

 

35.6.3.1.2 - Separable Equations

• In a separable equation the differential can be split so that it is on both sides of the equation. We then integrate to get the solution. This typically means there is only a single derivative term.

 

35.6.3.1.3 - Homogeneous Equations and Substitution

• These techniques depend upon finding some combination of the variables in the equation that can be replaced with another variable to simplify the equation. This technique requires a bit of guessing about what to substitute for, and when it is to be applied.

 

35.6.3.2 - Second-order Differential Equations

• These equations have at least one second-order derivative.

• In engineering we will encounter a number of forms,

- homogeneous

- nonhomogeneous

35.6.3.2.1 - Linear Homogeneous

• These equations will have a standard form,

 

• An example of a solution is,

 

35.6.3.2.2 - Nonhomogeneous Linear Equations

• These equations have the general form,

 

• to solve these equations we need to find the homogeneous and particular solutions and then add the two solutions.

 

• Consider the example below,

 

35.6.3.3 - Higher Order Differential Equations
35.6.3.4 - Partial Differential Equations

• Partial difference equations become critical with many engineering applications involving flows, etc.

35.6.4 Other Calculus Stuff

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• The Taylor series expansion can be used to find polynomial approximations of functions.

 

35.6.5 Practice Problems

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1. Find the derivative of the function below with respect to time.

 

2. Solve the following differential equation, given the initial conditions at t=0s.

 

3. Find the following derivatives.

 

4. Find the following integrals

 

5. Find the following derivative.

6. Find the following derivatives.

 

7. Solve the following integrals.

 

8. Solve the following differential equation.

 

9. Set up an integral and solve it to find the volume inside the shape below. The shape is basically a cone with the top cut off.

 

 

10. Solve the first order non-homogeneous differential equation below. Assume the system starts at rest.

11. Solve the second order non-homogeneous differential equation below.

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