Slopes and Areas

Slopes and Areas

• Frequently we will want to know the slope of a curve at some point.

• Or an area under a curve.

We calculate area under a curve as the sum of areas of many rectangles under the curve.

We calculate slope as the change in height of a curve during some small change in horizontal position: i.e. rise over run

Review: Axes

• When two things vary, it helps to draw a

picture with two perpendicular axes to show what they do. Here are some examples:

y

x

x

t

y varies with x x varies with t

Here we say “ y is a function of x” . Here we say “x is a function of t” .

Positions

• We identify places with numbers on the axes

The axes are number lines that are perpendicular to each other.

Positive x to the right of the origin (x=0, y=0), positive y above the origin.

Straight Lines

• Sometimes we can write an equation for how one variable varies with the other. For example a straight line can be described as

y = ax + b Here, y is a position on the

line along the y-axis, x is a position on the line along the x-

axis, a is the slope, and b is the

place where the line hits the y-axis

Straight Line Slope

y = ax + b The slope, a, is just the rise y divided by the run x. We can do this anywhere on the line.

So the slope of the line here is y = ^-3

x Remember: Rise over Run² and up and right are positive

Proceed in the positive x direction for some number of units, and count the number of

units up or down the y changes

y- intercept

y = ax + b is our equation for a line

b is the place where the line hits the y-axis

The intercept b is y = +3 when x = 0 for this line

We want an equation for this line

y = ax + b is the general equation for a line

So the equation of the line here is y =

-

2

Equation of our example line

We plugged in the slope and y intercept

An example: a PT diagram for 2 phases

• Suppose we plot the boundary between the stable PT conditions for two minerals

At, for example, 1 GPa and T= 300K, Phase 2 is stable

• • Suppose a piston moves up due to expansion of a confined gas

• • V1 = 1000 cm3, under a confining pressure of 10 bars

• • V2 = 2000 cm3, pressure relaxes to 5 bars

• • Work is done…

Another example

Trig

• Perpendicular axes and lines are very handy. Recall we said we use them for vectors such as velocity. To break a vector r into

components, we use trig. The rise is r ^.. sin and the run is r cos 

This vector with size r and direction , has been broken down into components.

Along the y-axis, the rise is y = +r sin  Along the x-axis, the run is x = +r cos 

Demo: The sine is the ordinate (rise) divided by the hypotenuse

sin  = rise / r so the rise = r sin 

Similarly the run = r cos 

hypotenuse

rise

run

Whenever possible we work with unit vectors so r = 1, simplifying calculations.

Okay, sines and cosines, but what’s a Tangent?

A Tangent Line is a line that is going in the

direction of a point proceeding along the curve.

A Tangent at a point is the slope of the curve there.

A tangent of an angle is the sine divided by the cosine.

Positive slopes shown in green, zero slopes are black, negative are red.

Tangents to curves

• Here the vector r vector r shows the velocity of an ion moving along the blue line f(x)

• At point P, the particle has speed the length of rr and the direction shown makes an angle  to the x-axis

slope = f(x + h) –f(x) (x + h) – x

This is rise over run as always Lets see that is r sin tan

r cos 

P

The slope is a tangent to the curve.

P

Slope at some point on a curve

• We can learn the same things from any curve if we have an equation for it.

We say y = some function f of x, written y = f(x). Lets look at the small interval between x and x+h. y is different for these two values of x.

The slope is rise over run as always slope = f(x + h) –f(x)

(x + h) – x

rise

derivative dy/dx = f(x + h) –f(x) lim h=>0 h

The exact slope at some point on the curve is found by making the distance between x and x+h small, by making h really small. We call it the derivative.

This is inaccurate for a point on a curve, because the slope varies.

run

A simple derivative for Polynomials

• The exact slope “derivative” of f(x)

f’(x) = f(x + h) – f(x) = f(x + h) – f(x) lim h=>0 (x + h) – x lim h=>0 h

is known for all of the types of functions we will use in Petrology.

For example, suppose y = x

where n is some constant and x is a variable Then y’(x) = dy/dx = nx

dy/dx means “The small change in y with respect to a small change in x”

Some Examples for Polynomials

• (1) Suppose y = x

. What is dy/dx?

dy/dx = 4x

• (2) Suppose y = x

What is dy/dx?

dy/dx = -2x

We just saw for polynomials y = xⁿ the dy/dx = nx^{n - 1}

Differentials

• Those new symbols dy/dx mean the really accurate slope of the function y = f(x) at any

point. We say they are algebraic, meaning dx and dy behave like any other variable you

manipulated in high school algebra class.

• The small change in y at some point on the

function (written dy) is a separate entity from dx.

• For example, if y = x

• dy/dx = nx

also means dy = nx

dx

Variable names

• There is nothing special about the letters we use except to remind us of the axes in our

coordinate system

• For example, if y = u

• dy = nu

du is the same as the previous formula.

y = uⁿ

u

Constants Alone

• The derivative of a constant is zero.

• If y = 17, dy/dx = 0 because constants don’t change, and the constant line has zero slope

Y = 17 17

y

x

For any dx, dy = 0

X alone

• Suppose y = x What is dy/dx?

• Y = x means y = x¹. Just follow the rule.

• Rule: if y = xⁿ then dy/dx = nx^{n – 1}

• So if y = x, dy/dx = 1x⁰ = 1

• Anything to the power zero is one.

A Constant times a Polynomial

• Suppose y = 4 x

What is dy/dx?

• Rule: The derivative of a constant times a polynomial is just the constant times the derivative of the polynomial.

• So if y = 4 x

, dy/dx = 4

( 7x

)

Multiple Terms in a sum

• The derivative of a function with more than one term is the sum of the individual

derivatives.

• If y = 3 + 2t + t

then dy/dt = 0 + 2 +2t

• Notice 2t = 2t

For polynomials y = xⁿ dy/dx = nx^{n - 1}

The derivative of a product

• In words, the derivative of a product of two

terms is the first term times the derivative of

the second, plus the second term times the

derivative of the first.

Exponents

• a

a

= a

a

/a

= a

• (a

)

= a

(ab)

= a

b

• (a/b)^m

= a

/b

a

= 1/a

Suppose m and n are rational numbers

You can remember all of these just by experimenting

For example 2² = 2x2 and 2⁴= 2x2x2x2 so 2²x2⁴ = 2x2x2x2x2x2 = 2⁶ reminds you of rule 1

Rule 6, a^-n = 1/aⁿ , is especially useful

Logarithms

• Logarithms (Logs) are just exponents

• if b^y = x then y = logb x

• log10 (100) = 2 because 10² = 100

• Natural logs (ln) use e = 2.718 as a base

• For example ln(1) = loge(1) = 0 because e⁰ = (2.718)⁰ = 1

Anything to the zero power is one.

e

• e is a base, the base of the so-called natural logarithms just mentioned. e ~ 2.718

• It has a very interesting derivative (slope).

• Suppose u is some function

• Then d(e

) = e

du

• “The derivative of e

is e

times the derivative of u”

• Example: If y = e

what is dy/dx?

• here u = 2x, so du = 2

• Therefore dy/dx = e

2

Integrals

• The area under a function between two values of, for

example, the horizontal axis is called the integral. It is a sum of a series of very tall and thin rectangles, and is indicated by a script S, like this:

•

Integrals

• To get accuracy with areas we use extremely

thin rectangles, much thinner than this.

Example 1

• If y=3x⁵ Then dy/dx = 15x⁴

• Then y = 15x⁴ dx = 3x⁵ + a constant

Integration is the inverse operation for differentiation

We have to add the constant as a reminder because, if a constant was present in the original function, it’s derivative would be zero and we wouldn’t see it.

Example2: a trick

Sometimes we must multiply by one to get a

known integral form. For example, we know:

A useful method

• When a function changes from having a

negative slope to a positive slope, or vs. versa, the derivative goes briefly through zero.

• We can find those places by calculating the

derivative and setting it to zero.

Getting useful numbers

• Suppose y = x².

• (a) Find the minimum

If y = x² then dy/dx = 2x¹ = 2x. Set this equal to zero 2x=0 so x=0

y = x² so if x = 0 then y = 0

Therefore the curve has zero slope at (0,0)

Getting useful numbers

• Suppose y = x².

• TODO: Find (a) the location of the minimum, and (b) the slope at x=3 (a)See previous page

(b) dy/dx = 2x , so set x=3

then the slope is 2x = 2 ^. 3 = 6

Getting useful numbers

• Here is a graph of y = x²

• Notice the slope is zero at (0,0), the minimum

• The slope at (x=3,y=9) is +6/1 = 6