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ITT Technical Institute
Framingham, Massachusetts
Quarter 7 Electronics Calculus
Instructor: Paul Burney

Review for Final Examination

Topics Covered:

• Finding Area by approximation with rectangles
• Finding limits as functions approach certain numbers
• Finding Rates of Change with the Derivative
• Using the Derivative for Applications
• Finding Anti-derivatives
• Finding Area under a curve with Integrals
• Using the integral for applications
• Area between curves
• Average values

Brief History of Calculus:

Calculus was invented concurrently by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. Calculus was used to study speed and motion and other topics of Newton's new physics, the behavior of functions as certain limits were approached, as well as problems of finding areas under curved shapes accurately.

The two historic problems of calculus are the rate of change problem and the area problem. Derivative calculus allows us to calculate the change of something for an infinitesimally small amount of time, thus giving us a very accurate instantaneous rate of change.

To calculate the area under a curve, it can be divided up into small rectangles whose areas can then be summed. This method can be somewhat improved by making the rectangles thinner or by using a midpoint approximation, but each of these methods involves intense computations. Integral calculus allows us to calculate the area under a curve by summing up infinitesimally small rectangles, thus finding a true area and not an approximation of area. It makes this possible via the fundamental theorem of calculus which relates integrals over a certain range to the anti-derivatives of the function over that range.

Mechanics of the Final

The test consists of 24 questions, 3 of which are multiple choice. The questions cover all of the topics above. About 40% of the problems involve derivatives, and the other 40% involve integrals, the remaining 20% involve the other topics.

Exam Practice Review

The following review covers most of the topics on the test. It is not in the same format as the test but the material is similar. The answers to the review are found in a separate document.

1. Find the limit of the following:

2. Find the derivative of the following using implicit differentiation:

3. Find the acceleration of a car at 0.25 seconds if its velocity in m/s is given by:

4. Find the area under the curve y = 3x2 from 1 to 5, using 4 intervals.
5. Find the following Integral:

6. Find the area between the two curves below:

7. A capacitor takes 30 ms to charge to full capacity of 43.5 mC. After t ms, the charge on the capacitor is given by:

What is the average charge (in mC) on the capacitor as it is charging? (Hint: you don't need to mess with units since the equation is in terms of mC and ms.)