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The Quick Guide to Calculus

The Quick Guide to Calculus. The derivative. Derivative. A derivative measures how much a function changes for various inputs of that function. It is like the instantaneous slope at any point on a function (and this can be complex or simple depending on the function).

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The Quick Guide to Calculus

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  1. The Quick Guide to Calculus

  2. The derivative

  3. Derivative A derivative measures how much a function changes for various inputs of that function. It is like the instantaneous slope at any point on a function (and this can be complex or simple depending on the function)

  4. What will the derivative look like? • y = a • dy/dx = ?? • dy/dx = 0

  5. What will the derivative look like? • y = mx • dy/dx = ?? • dy/dx = m

  6. What will the derivative look like? • y = x2 • dy/dx = ?? • dy/dx = 2x

  7. examples Can you match the graphs on the left to their derivative functions on the right? 1 a b a d c 1 ____ 2 ____ 3 ____ 4 ____ 2 b 3 c 4 d

  8. Now let’s look at it mathematically

  9. The “Power Rule”

  10. Other Important Rules But also, from the power rule:

  11. Other Important Rules

  12. Now YOU try it Determine the derivatives of the following functions

  13. 1. y = x3 4. y = 4 dy/dx = 0 dy/dx = 3x2 2. y = 2x2 5. y =x-4 y’ = 4x y’ = -4(x)-5 6. y = ½ x1/2 3. y = 3x4 – 8x d/dx (y) =1/4 x-1/2 d/dx (y) = 12x3 – 8

  14. Integrals: The ANTI Derivative • An integral is opposite of a derivative • If 2x is the derivative of x2, then x2 is the integral (or anti-derivative) of 2x • What would the integral of of 4x3 be? • x4

  15. Integral: The Area Under A Curve The area under a curve can be found by dividing the whole area into tiny rectangles of a finite width and a height equal to the value of the function at the center of each rectangle This becomes more precise the smaller you make the rectangles Then you add up all the rectangles

  16. Integral: The Area Under a Curve The approximation to the area becomes better as the rectangles become smaller (N∞,Δx0) and this is what an integral is:

  17. Integral: Some examples • For a function that is just a constant, a, then the area under the curve would be a rectangle: • For a linear function f(x)=ax, the area under the curve would be a triangle:

  18. Integral: The Anti-Derivative The general equation for the integral: Remember that for a derivative it was: (So the equation for the integral should make sense, it’s the anti-derivative)

  19. Now YOU try it Determine the Integrals of the following functions

  20. 1. f(x) = 6x5 4. y = 4 2. f(x) = -6x-7 5. f(x) =½x-½ 6. y = ½ x1/2 3. y = 10x4 + x

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