Slopes and Areas

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 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 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 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

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. Dy means y finish – y start, here 0 - 3 = -3 Dx means x finish – x start, here 2 - 0 = +2 Or, 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 Dx 2 Remember: Rise over Run and up and right are positive

6. y- intercept y = ax + b The intercept b is y = +3 when x = 0 for this line

7. Equation for this line y = ax + b Equation of Example Line So the equation of the line here is y =-3 x + 3 2

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 cloudburst caused the creek to flow at 3 m3/hour initially, but always decreased at a rate (slope) of - 3/2 m3 per hour after that, so it stopped after two hours. The area under the line is the total volume of water the flowed past the gauge during the two hours. A = 1/2bh = 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 into components, we use trig. The sine of angle theta is r times the vertical (rise) part of this triangle, and the cosine of angle is r times the horizontal (run) . 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 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

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.

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 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 The slope, and by extension the accurate derivative with h very small, 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 derivative dy/dx = f(x + h) –f(x) lim h=>0 h

13. A simple derivative for Polynomials • The 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 Hydrology. For example, suppose y = xn where n is some constant and x is a variable Then dy/dx = nxn-1 dy/dx means “The change in y with respect to x”

14. For polynomials y = xn 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

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 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-Idx

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

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? • 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 = 0 + 2 +2t • Notice 2t1 = 2t

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 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

24. e • e is a base, the base of the so-called natural logarithms. • It has a very interesting derivative. • Suppose u is some function • Then d(eu) = eu du • 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 small rectangles, and is indicated by a very tall and thin script S, like this:

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

27. 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.

28. Example2: a trick Sometimes we must multiply by one to get a known integral form. For example, we know:

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. • (b) Find the slope at x=3 (a) If y = x2 then dy/dx = 2x1 = 2x. Set x=3 then the slope is 2x = 2 . 3 = 6

32. Getting useful numbers • Here is a graph of y = x2 • Notice the slope is zero at (0,0) • The slope at (x=3,y=9) is +6/1 = 6