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2413 Calculus I Chapter 4(1) Anti Derivatives

2413 Calculus I Chapter 4(1) Anti Derivatives. Take each derivative. Steps to take Derivative. 1) Make a polynomial (no fractions). 2) Multiply the power by the front number. 3) Subtract one from the power. 4) Simplify the answer.

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2413 Calculus I Chapter 4(1) Anti Derivatives

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  1. 2413 Calculus IChapter 4(1)Anti Derivatives

  2. Take each derivative

  3. Steps to take Derivative 1) Make a polynomial (no fractions) 2) Multiply the power by the front number 3) Subtract one from the power 4) Simplify the answer

  4. Reversing the process gives the anti-derivative (the Integral) 1) Make a polynomial (no fractions) 2) Add one to the power 3) Divide by the new power 4) Simplify the answer

  5. Symbols used: Derivative from chain rule Antiderivative After years of finding mathematics easy, I finally reached integral calculus and came up against a barrier. I realized that this was as far as I could go, and to this day I have never successfully gone beyond it in any but the most superficial way." -- Isaac Asimov

  6. Example 1 Add 1 to the power Divide by the power Simplify

  7. Check the Answer What about adding a constant? This is Correct! This is also Correct! There are an infinite number of Antiderivatives for a given function, each is the same except for a constant.

  8. All Steps to Integrate 1) Make a polynomial (no fractions) 2) Add one to the power 3) Divide by the new power 4) Simplify the answer 5) Add a constant to the end + C

  9. Example 2 Make a polynomial Add 1 to power Divide by Power Simplify and add C

  10. Trig Integrals

  11. e and Ln Integrals

  12. Example 3 Use the rules for each function Simplify and add C

  13. Initial Value Problems – finding C You can find C if you are given a point on the curve or some value for the function. 1) Integrate the function 2) Put in the point or value you are given 3) Solve for C

  14. Example 4 Integrate Put in the point and solve for C Write equation without C

  15. A ball is thrown upward from the top of a 336 ft. building with an initial velocity of 320 ft/sec. Find the equation for the position at any time t. Example 5 Integrate Constant Integrate Constant This is the equation for position.

  16. At what time does the ball reach the maximum height? How long does it take to hit the ground? Example 6 At max height Velocity = 0 At the ground position = 0

  17. Differential Equations(Equations that contain a derivative) There are entire courses devoted to solving differential equations: Ordinary Differential Equations (ODE’s) Partial Differential Equations (PDE’s)

  18. Example 7. Solve the differential equation Integrate Put in (0,6) and find C Integrate Put in (0,3) and find C

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