The Quick Guide to Calculus. The derivative Derivative A derivative measures how much a function changes for various inputs of that function. It is.

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Presentation transcript:

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)

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

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

What will the derivative look like? y = x 2 dy/dx = ?? dy/dx = 2x

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

Now let’s look at it mathematically

The “Power Rule”

Other Important Rules But also, from the power rule:

Other Important Rules

Now YOU try it Determine the derivatives of the following functions

1. y = x 3 dy/dx = 3x 2 2. y = 2x 2 y’ = 4x 3. y = 3x 4 – 8x d/dx (y) = 12x 3 – 8 4. y = 4 dy/dx = 0 5. y =x -4 y’ = -4(x) y = ½ x 1/2 d/dx (y) =1/4 x -1/2

Integrals: The ANTI Derivative An integral is opposite of a derivative If 2x is the derivative of x 2, then x 2 is the integral (or anti-derivative) of 2x What would the integral of of 4x 3 be? x 4

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

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

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:

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 antiderivative)

Now YOU try it Determine the Integrals of the following functions

1. f(x) = 6x 5 2. f(x) = -6x y = 10x 4 + x 4. y = 4 5. f(x) =½x -½ 6. y = ½ x 1/2