Slopes and Areas Frequently we will want to

Slopes and Areas • Frequently we will want to know the slope of a curve at some point. We calculate slope as the change in height of a curve during some small change in horizontal position: i. e. rise over run • Or an area under a curve. We calculate area under a curve as the sum of areas of many rectangles under the curve.

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 y varies with x Here we say “ y is a function of x”. t x varies with t 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 xaxis, 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 Dy divided by the run Dx. We can do this anywhere on the line. Proceed in the positive x direction for some number of units, and count the number of units up or down the y changes So the slope of the line here is Dy = -3 2 Dx Remember: Rise over Run and up and right are positive

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

y = ax + b is the general equation for a line We want an equation for this line Equation of our example line So the equation of the line here is y = -3 x + 3 2 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= 300 K, Phase 2 is stable

• • • Suppose a piston moves up due to expansion of a confined gas • V 1 = 1000 cm 3, under a confining pressure of 10 bars • V 2 = 2000 cm 3, 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 q, and the run is r cos q. Demo: The sine is the ordinate (rise) divided by the hypotenuse sin q = rise / r so the rise = r sin q Similarly the run = r cos q hy run rise se u en t po This vector with size r and direction q, has been broken down into components. Along the y-axis, the rise is Dy = +r sin q Along the x-axis, the run is Dx = +r cos q 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 shows the velocity of an ion moving along the blue line f(x) • At point P, the particle has speed the length of r and the direction shown makes an angle q 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 q r cos q PP = tan q The slope is a tangent to the curve.

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 rise run slope = f(x + h) –f(x) (x + h) – x This is inaccurate for a point on a curve, because the slope varies. 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 dy/dx = f(x + h) –f(x) lim h=>0 h

A simple derivative for Polynomials • The exact slope “derivative” of f(x) 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 = xn where n is some constant and x is a variable Then y’(x) = dy/dx = nxn-1 dy/dx means “The small change in y with respect to a small change in x”

We just saw for polynomials y = xn the dy/dx = nxn - 1 Some Examples for Polynomials • (1) Suppose y = x 4. What is dy/dx? dy/dx = 4 x 3 • (2) Suppose y = x-2 What is dy/dx? dy/dx = -2 x-3

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 = xn • dy/dx = nxn-I also means dy = nxn-I 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 = un • dy = nun-I 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 Y = 17 x For any dx, dy = 0

X alone • Suppose y = x What is dy/dx? • Y = x means y = x 1. Just follow the rule. • Rule: if y = xn then dy/dx = nxn – 1 • So if y = x , dy/dx = 1 x 0 = 1 • Anything to the power zero is one.

A Constant times a Polynomial • Suppose y = 4 x 7 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 7 , dy/dx = 4. ( 7 x 6)

For polynomials y = xn dy/dx = nxn - 1 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 + 2 t + t 2 then dy/dt = 0 + 2 +2 t • Notice 2 t = 2 t 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 Suppose m and n are rational numbers • aman = am+n • (am)n = amn • (a/b)m = am/bm am/an = am-n (ab)m = ambm a-n = 1/an You can remember all of these just by experimenting For example 22 = 2 x 2 and 24= 2 x 2 x 2 x 2 so 22 x 24 = 2 x 2 x 2 x 2 = 26 reminds you of rule 1 Rule 6, a-n = 1/an , is especially useful

Logarithms • Logarithms (Logs) are just exponents • if by = x then y = logb x • log 10 (100) = 2 because 102 = 100 • Natural logs (ln) use e = 2. 718 as a base • For example ln(1) = loge(1) = 0 because e 0 = (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(eu) = eu du • “The derivative of eu is eu times the derivative of u” • Example: If y = e 2 x what is dy/dx? • here u = 2 x, so du = 2 • Therefore dy/dx = e 2 x. 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 Integration is the inverse operation for differentiation • If y=3 x 5 Then dy/dx = 15 x 4 • Then y = 15 x 4 dx = 3 x 5 + a constant 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.

Example 2: 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 = x 2. • (a) Find the minimum If y = x 2 then dy/dx = 2 x 1 = 2 x. Set this equal to zero 2 x=0 so x=0 y = x 2 so if x = 0 then y = 0 Therefore the curve has zero slope at (0, 0)

Getting useful numbers • Suppose y = x 2. • TODO: Find (a) the location of the minimum, and (b) the slope at x=3 (a) See previous page (b) dy/dx = 2 x , so set x=3 then the slope is 2 x = 2. 3 = 6

Getting useful numbers • Here is a graph of y = x 2 • Notice the slope is zero at (0, 0), the minimum • The slope at (x=3, y=9) is +6/1 = 6