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 = x2 dy/dx = ?? dy/dx = 2x

examplesCan you match the graphs on the left to their derivative functions on the right?

1

2

3

4

a

b

c

d

1 ____

2 ____

3 ____

4 ____

b

a

d

c

Now let’s look at it mathematically

The “Power Rule”

Other Important Rules

But also, from the power rule:

d

dxx n nx n 1 d

dxc d

dxcx 0 0 cx 1 0

Other Important Rules

Now YOU try it

Determine the derivatives of the following functions

1. y = x3

dy/dx = 3x2

2. y = 2x2

y’ = 4x

3. y = 3x4 – 8x

d/dx (y) = 12x3 – 8

4. y = 4

dy/dx = 0

5. y =x-4

y’ = -4(x)-5

6. y = ½ x1/2

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 x2, then x2 is the integral (or anti-derivative) of 2x

What would the integral of of 4x3 be? x4

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∞,Δx0) 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

x ndx x n1

n 1C

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)

d

dxx n nx n 1

Now YOU try it

Determine the Integrals of the following functions

1. f(x) = 6x5

2. f(x) = -6x-7

3. y = 10x4 + x

4. y = 4

5. f(x) =½x-½

6. y = ½ x1/2

f (x)dx x 6

ydy 4x

f (x)dx x 6

ydy 2x 5 12 x

2

f (x)dx x12

f (x)dx 13 x

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