1. 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 slope as the change in height of a curve during some small change in horizontal position: i.e. rise over run
We calculate area under a curve as the sum of areas of many rectangles under the curve.

2. 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
We say “ y is a function of x” or “x is a function of t”
x varies with t
y varies with x
Here we say “ y is a function of x” Here we say “x is a function of t” .

3. Positions We identify places with numbers on the axes
Each tick mark on the axes is one away from its neighbor. 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.
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.

4. 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
Notice here that as x increases, y decreases

5. 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
Notice here that as x (the run) increases, y (the rise) decreases. This line has a negative slope.
So the slope of the line here is Dy = Dx
Remember: Rise over Run
and up and right are positive

6. 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
Each point on the line has position (x,y). The line has a negative slope -3/2 and the y value where x=0 (the intercept) is +3

7. We want an equation 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
Each point on the line has position (x,y). The line has a negative slope -3/2 and the y value where x=0 (the intercept) is +3
We plugged in the slope and y intercept

8. An example: a flow gauge on a small creek
Suppose we plot as the vertical axis the flow rate in m3/ hour and the horizontal axis as the time in hours
Then the line tells us that a flash flood caused the creek to flow at 3 m3/hour initially, but flow decreased at a rate (slope) of - 3/2 m3 per hour after that, so it stopped after two hours.
BTW, the area under the line tells us the total volume of water the flowed past the gauge during the two hours.
Area of a triangle = 1/2bh
Area = 1/2 x 2hr x 3m3/hr = 3 m3
Area of a triangle is half of a rectange A = 1/2 base x height = 1/2 x 2 x 3 = 3 m3
This plot, flow vs. time, is a hydrograph. The area under the curve is the volume of runoff.

9. 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
Whenever possible we work with unit vectors so r = 1, simplifying calculations.
hypotenuse
rise
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
run
Whenever possible we work with unit vectors so r = 1, simplifying calculations.

10. 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.

11. Tangents to curves
Here the vector r shows the velocity of a particle 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 = tan q
r cos q
But the sine over the cosine is called the tangent
The slope is a tangent to the curve. P

12. 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
This is inaccurate for a point on a curve, because the slope varies.
run
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.
Derivative is the same equation as slope except that h gets very small. In the slope equation’s denominator the x and –x add to zero
derivative dy/dx = f(x + h) –f(x)
lim h=> h

13. 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=> (x + h) – x lim h=> h
is known for all of the types of functions we will use in Hydrology.
For example, suppose y = xn
where n is some constant and x is a variable
Then y’(x) = dy/dx = nxn-1
dy/dx “The change in y wrt x”
dy/dx means “The small change in y with respect to a small change in x”

14. Some Examples for Polynomials
We just saw for polynomials y = xn the dy/dx = nxn - 1
Some Examples for Polynomials
(1) Suppose y = x4 . What is dy/dx?
dy/dx = 4x3
(2) Suppose y = x-2
What is dy/dx?
dy/dx = -2x-3
dy/dx = nxn-1

15. 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

16. 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

17. 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
Y = 17 17 x
For any dx, dy = 0

18. 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 = nxn – 1
So if y = x , dy/dx = 1x0 = 1
Anything to the power zero is one.

19. A Constant times a Polynomial
Suppose y = 4 x7 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 x7 , dy/dx = 4 . ( 7x6)

20. 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 + 2t + t2 then dy/dt = t
Notice 2t = 2t1

21. 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.

22. Exponents aman = am+n am/an = am-n (am)n = amn (ab)m = ambm
Suppose m and n are rational numbers
aman = am+n am/an = am-n
(am)n = amn (ab)m = ambm
(a/b)m = am/bm a-n = 1/an
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

23. Logarithms Logarithms (Logs) are just exponents
if by = x then y = logb x
log10 (100) = 2 because 102 = 100
Natural logs (ln) use e = as a base
For example ln(1) = loge(1) = 0
because e0 = (2.718)0 = 1
Anything to the zero power is one.

24. 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 = e2x what is dy/dx?
here u = 2x, so du = 2
Therefore dy/dx = e2x . 2

25. 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:

26. Integrals
To get accuracy with areas we use extremely thin rectangles, much thinner than this.

27. Example 1 Integration is the inverse operation for differentiation
If y=3x5 Then dy/dx = 15x4
Then y = x4 dx = 3x5 + 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.

28. Example2: a trick
Sometimes we must multiply by one to get a known integral form. For example, we know:
Remember d(eu) = eu du. If we find an integral in this form, we know the answer is the original function before differentiation.
Here multiplying by a/a = 1 puts the integral in the right form.
Note the constant. We can evaluate a derivative easily. Integrals are not all known.

29. 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.

30. 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)

31. Getting useful numbers
Suppose y = x2.
TODO: Find (a) the location of the minimum, and (b) the slope at x=3
See previous page
(b) dy/dx = 2x , so set x=3
then the slope is 2x = = 6

32. 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