Tangent to a Curve and Derivative of a Functionmrswoodley.weebly.com/uploads/5/9/1/8/5918762/jw_tangent...1 Tangent to a Curve and Derivative of a Function So exactly what is this

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Tangent to a Curve and Derivative of a Function

So exactly what is this 'Calculus' thing?

Calculus is a set of techniques developed for two main reasons:1) finding the gradient at any point on a curve, and2) finding the area enclosed by curved boundaries.

Archimedes (2nd Century BC), Newton and Leibniz are the namesof Mathematicians over time who have contributed to building an understanding of the ideas of Calculus.

Areas in which Calculus is used include science, engineering, and medicine.

Gottfried Wilhelm LeibnizArchimedes

Board of Studies

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GradientYou should know gradient is defined as rise/run. Up until now, you have only found the gradient of straight lines. The gradient also measures the rate of change of the dependent variable with respect to the rate of change in the independent variable.

At any point on a curve, the gradient of the curve (at that point)can be approximated by the gradient of the tangent at that point. Look at the geogebra file below:

geogebra activitydemonstration

Like that of a line, the gradient of a curve can be positive, negative or zero, depending on which part of the curve you are referring to.

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Drag out the symbols to show where the gradient of each curve is positive, negative or zero.

+_0

drag them

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Graphing the Gradient Function

Where the gradient of a curve f(x)is positive, we draw the gradient function above the x axis. Where the gradient of f(x) is negative, we draw the gradient function below the x axis. Where the gradient of the curve is zero, the gradient function crosses the x axis.

online activityhttp://www.math.uri.edu/~bkaskosz/flashmo/derplot/

http://www.math.uri.edu/~bkaskosz/flashmo/derplot/

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Try this one:

online activityderivative grapher

http://webspace.ship.edu/msrenault/GeoGebraCalculus/derivative_try_to_graph.html

http://webspace.ship.edu/msrenault/geogebracalculus/derivative_try_to_graph.html

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Groves

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A function is called a differentiable function if the gradient of the tangent can be found.There are some graphs that are not differentiable in places.Most functions are continuous , which means that they have a smooth unbroken line or curve. However, some have a gap, or discontinuity, in the graph (e.g. hyperbola). This can be shown by an asymptote or a ‘hole’ in the graph. We cannot find the gradient of a tangent to the curve at a point that doesn’t exist! So the function is not differentiable at the point of discontinuity.

DifferentiabilityThe process of finding the gradient of a tangent is called differentiation .The resulting function is called the derivative .

This function has a 'gap' at x=a, so is not differentiable at when x=a.

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Discuss the differentiability of these functions:

rub and reveal

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Groves

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A mathematician must know his/her limits.

(Old Chinese Proverb)

Limits

Consider the two equations and graphs below:1) y = x+22) y = x24 =(x2)(x+2) = x+2

x2 (x2)

Are they the same?

rub and reveal

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This curve is discontinuous at x=2

So how does the concept of a limit work?Try this:1) Substitute in x=2 and see what happenslim x24 = 224

x2 22= 0 0

very bad!!!

2) Instead, factorise, simplify and THEN substitutelim x24 = lim (x2)(x+2)

x2 (x2)

x 2

x 2

x 2

x 2

= lim (x+2)

= 4

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Important Definition:

Furthermore, a function f(x) is called continuous or a continuous function if it is continuous at each point in its domain, ie if f(x) is continuous at x = c for every choice of c in the domain of the function.

We say a function is continuous if:(i) f(x) is defined at c;(ii) the limit of f(x) as x approaches c exists;(iii) f(c) is equal to this limit.

We use the notationslim f(x), and lim f(c + h)

h 0x c

to mean the limit of the function as x c.

If f(x) is ‘continuous’ at x = c, then lim f(x) = f(c), and lim f(c + h) = f(c), for negative and positive values of h.

x c

h 0

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To differentiate from first principles, we need to look more closely at the concept of a limit.A limit is used when we want to move as close as we can to something.Often this is to find out where a function is near a gap or discontinuous point.

Using Limits

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Groves

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To differentiate from first principles , we first use the point of contact and another point close to it on the curve (this line is called a secant ) and then we move the second point closer and closer to the point of contact until they overlap and the line is at single point (the tangent ). To do this, we use a limit.Using geogebra, sketch a parabola, then zoom in and observe the appearance of the curve as you magnify it more.

Differentiating by First Principles

geogebra activity

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We can find a general formula for differentiating from first principles by using c rather than any particular number. We use general points P(c,f(c)) and Q(x,f(x)) where x is close to c .The gradient of the secant PQ is given by

m = f(x)f(c) xc

General Principles

The gradient of the tangent at P is found when x approaches c. We call this f'(c) .

There are other versions of this formula.We can call the points P(x, f(x)) and Q((x+h), f(x+h)) where h is small.Secant PQ has gradient:

m = y2y1 x2x1

= f(x+h) f(x) x+hx

= f(x+h)f(x) h

continued next slide

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As h approaches 0, the gradient of the tangent becomeslim f(x+h)f(x) We call this f'(x) h

h 0

Because so many Mathematicians have contributed to the development of Calculus over time, there are many different notations. All of these different notations stand for the derivative, or the gradient of the tangent:

They all mean the same thing, but should reflect the way the original function is stated.We can also find the normal. The normal is the line perpendicular to the tangent at the point of contact.

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Differentiate from first principles to find the gradient of the tangent to the curve y = x2+3 at the point where x = 1.let y = f(x)f'(x) = lim f(x+h)f(x)

hf(x) = x2+3f(x+h) = (x+h)

2+3

f'(x) = lim (x+h)

2+3 (x

2+3)

h= lim (x

2+2hx+h

2)+3x

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h= lim (2hx+h

2)

h= lim h(2x+h)

h= lim (2x+h)

=2

h 0

h 0

h 0

h 0

h 0

h 0

2

2

2

2

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Now substitute in x=1

f'(1)

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Using the Derivative

Once you find the derivative of a function, you can use this to find the gradient of the tangent and normal at this point, and also to find the equation of these lines.

rub and reveal

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Groves Ex 8.4

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Fitzpatrick

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Groves

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But it takes so looong.....Mathematicians look at patterns to develop rules. Consider the examples below and see if you can work out a 'short' way to differentiate expressions.

f(x) = x5 f'(x) = 5x4

f(x) = x4 f'(x) = 4x3

f(x) = x3 f'(x) = 3x2

f(x) = 10x3 f'(x) = 30x2

f(x) = 5x8 f'(x) = 40x7

brainstorm

Rule:

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Groves

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What to do with harder questions?Expanding out brackets can be a technique for differentiating; but there are three other ways of finding a derivative:

Product Rule

Differentiating the product of two functions y = uv gives the result:

It is easier to remember this rule as y' = uv' + vu' . Some people also say 'first times derivative of the second plus second times derivative of first'). We can also write this the other way around which helps when learning the quotient rule.

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rub and reveal

Example of product rule

ALWAYS ensure you label 'u' and 'v' before you do anything else!

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Differentiating the quotient of two functions y = u gives the result: v

Quotient Rule

This can also be written as:

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Example of using quotient rule:

rub and reveal

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Composite Function RuleWhen you have a function which is a combination of other functions, you need to use the composite function (sometimes called the chain rule or function of a function rule).

example of using composite function rule:

rub and reveal

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Fitzpatrick

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Groves

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At the end of this topic:• Make sure you understand 'differentiable', limits, asymptotes etc• Be able to draw the gradient function for a given graph• Be able to differentiate by first principles• Use the rules for differentiation correctly• Ensure you can find the equations of tangents and normals to curves at a given point• Develop a topic summary

watch a movieCalculus Rhapsody

Attachments

Curves1stand2ndDerivative IWB.ggb

gradient demo.ggb

9507cac2ca433e3eb63f1ec0724654a9/ITC.gif

d90d49747e64eadf7660d490a956930f/LBCK2.png

geogebra_thumbnail.png

geogebra_javascript.js

function ggbOnInit() {}

geogebra.xml

SMART Notebook

geogebra_thumbnail.png

geogebra_javascript.js

function ggbOnInit() {}

geogebra.xml

SMART Notebook

Page 1: backgroundPage 2: gradientPage 3: Feb 2-3:50 PMPage 4: graphing gradient functionPage 5: Feb 2-3:50 PMPage 6: Feb 2-6:05 PMPage 7: Apr 14-3:43 PMPage 8: differentiabilityPage 9: Feb 2-3:50 PMPage 10: Feb 2-6:29 PMPage 11: limitsPage 12: Feb 2-3:50 PMPage 13: continuityPage 14: Feb 2-3:50 PMPage 15: Feb 3-2:43 PMPage 16: Feb 2-3:50 PMPage 17: Feb 2-3:50 PMPage 18: general principlesPage 19: Feb 2-3:50 PMPage 20: Feb 2-3:50 PMPage 21: Feb 2-3:50 PMPage 22: Feb 2-3:50 PMPage 23: Feb 3-5:58 PMPage 24: Apr 14-3:45 PMPage 25: Feb 3-6:00 PMPage 26: Apr 14-3:45 PMPage 27: Feb 3-2:46 PMPage 28: Feb 3-5:45 PMPage 29: Apr 14-3:46 PMPage 30: Feb 3-6:42 PMPage 31: Apr 14-3:46 PMPage 32: short methodPage 33: Feb 3-6:16 PMPage 34: Apr 14-3:46 PMPage 35: product rulePage 36: Feb 2-3:50 PMPage 37: quotient rulePage 38: Feb 2-3:50 PMPage 39: Feb 2-3:50 PMPage 40: Feb 3-7:09 PMPage 41: Feb 3-9:27 PMPage 42: Apr 14-3:47 PMPage 43: Feb 3-9:30 PMPage 44: Apr 14-3:47 PMPage 45: quotient ruleAttachments Page 1

Documents

Tangent to a Curve and Derivative of a Functionmrswoodley.weebly.com/uploads/5/9/1/8/5918762/jw_tangent...1 Tangent to a Curve and Derivative of a Function So exactly what is this