slopes and areas
DESCRIPTION
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. Review: Axes. - PowerPoint PPT PresentationTRANSCRIPT
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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.
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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” .
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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.
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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
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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.
y means y finish – y start, here 0 - 3 = -3
x means x finish – x start, here 2 - 0 = +2
So the slope of the line here
is y = -3
x 2Remember: Rise over Run and up and right are positive
Or, proceed in the positive x direction for some number of units, and count the number of units up or down the y changes
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y- intercept
y = ax + b The intercept b is y = +3 when x = 0 for this line
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Equation for this line
y = ax + b
So the equation of the line
here is y = -3 x + 3
2
Equation of Example Line
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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.
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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) .
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 hypotenusesin = rise / r so the rise = r sin
Similarly the run = r cos
hypo
tenus
e
rise
run
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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.
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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 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, and by extension the accurate derivative with h very small, is a tangent to the curve.
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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
This is inaccurate for a point on a curve, because the slope varies.run
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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 variableThen dy/dx = nxn-1
dy/dx means “The change in y with respect to x”
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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
For polynomials y = xn dy/dx = nxn - 1
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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-I dx
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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
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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 = 1717
y
x
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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.
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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)
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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
For polynomials y = xn dy/dx = nxn - 1
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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.
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Exponents
• aman = am+n am/an = am-n
• (am)n = amn (ab)m = ambm
• (a/b)m = am/bm a-n = 1/an
Suppose m and n are rational numbers
You can remember all of these just by experimentingFor example 22 = 2x2 and 24= 2x2x2x2 so 22x24 = 2x2x2x2x2x2 = 26
reminds you of rule 1Rule 6, a-n = 1/an , is especially useful
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Logarithms
• Logarithms (Logs) are just exponents
• if by = x then y = logb x
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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
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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:
•
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Integrals
• To get accuracy with areas we use extremely thin rectangles, much thinner than this.
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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.
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Example2: a trickSometimes we must multiply by one to get a known integral form. For example, we know:
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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.
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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)
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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
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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