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Preview of Calculus

AP Calculus AB Limits. Preview of Calculus. AP Calculus AB Functions. Why?. Precalculus mathematics is static (fixed). You can only analyze an object at a single point in time. . You cannot analyze an object that is moving.

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Preview of Calculus

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  1. AP Calculus AB Limits Preview of Calculus AP Calculus AB Functions

  2. Why? Precalculus mathematics is static (fixed). You can only analyze an object at a single point in time. You cannot analyze an object that is moving. The problem with that is that most things in the real world are not static, they move.

  3. Without Calculus With Calculus Turn to page 43 in your textbook.

  4. Who? Where? When? Abu Ali al-Hasanibn al-Haytham Basra, Persia (Iraq) around 970 B.C. Archimedes Ancient Greece around 250 B.C. Sir Isaac Newton England 1643-1727 Gottfried Wilhelm von Leibniz Germany 1646-1716 Aryabhata Patna (India) around 500 A.D. Abu Bakr al-Karaji Baghdad, Persia (Iraq) around 1000 A.D. Jia Xian China around 1000 A.D. https://www.youtube.com/watch?v=axZTv5YJssA

  5. What?

  6. What is Calculus? Calculus is the mathematics of change. It allows us to take an infinite number of points that are nearly zero in value and make some sense of the overall change that occurring. 2 Basic Types Differential Integral Rate of Change Area Volume Rate of Change Velocity Acceleration https://www.youtube.com/watch?v=ismnD_QHKkQ

  7. How? TangentLine Problem When an object is in motion, its path is rarely in a straight line. We can tell a lot about the object by looking at its behavior at specific points. The slope of the tangent line at a single point allows us to see whether the function is increasing or decreasing and how quickly among other things. The problem is that since we only know one point on the line, we cannot find its slope using precalculus methods. file:///F:/2013-2014/Calculus/Tangent_Line_Problem.html

  8. How? Area Problem It is often necessary to find the area of a figure that has a curved surface. The problem is that geometric formulas cannot be used to obtain an exact area under a curve.

  9. The Limit After many years of studying these problems (among others) and investigating possible ways to solve them, mathematicians noticed that they could almost always find a “trend” in their results. As a secant line became closer and closer to a tangent line, the slope always approached a single number. As the area under a curve was calculated with more and more accuracy, that area always approached a single number. This number came to be called the limit of the data.

  10. The “BIG 3” of Calculus There are 3 basic ways to approach a problem in mathematics. 1. Numerically 2. Graphically 3. Analytically

  11. Decide whether the problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using calculus, solve it. If the problem seems to require calculus, explain your reasoning and use a graphical or numerical approach to estimate the solution. A family moves to a subdivision. They buy a lot to build a house on that is shaped like a parallelogram with side lengths of 105 feet and 160 feet. Find the area of their property to the nearest square foot.

  12. A family moves to a subdivision with lots built along a river. They buy a lot to build a house on that is shaped like a rectangle with the exception of the fourth side which is the river frontage. The front of the property is 160 feet across. The shortest distance from the front of the property to the river is 90 feet and the longest is 115 feet. Find the area of their property to the nearest square foot. Assignment p. 47: 1 – 11

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