So exactly what is this 'Calculus' thing?

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Leona Park
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finding the gradient at any point on a curve, and 2) finding the area

Archimedes (2nd Century BC), Newton and Leibniz are the names of

1 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, and 2) finding the area enclosed by curved boundaries. Archimedes (2nd Century BC), Newton and Leibniz are the names of 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. Tangent to a Curve and Derivative of a Function Archimedes Gottfried Wilhelm Leibniz Board of Studies 1

2 Gradient You 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: Tangent to a Curve and Derivative of a Function geogebra activity demonstration 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. 2

3 Drag out the symbols to show where the gradient of each curve is positive, negative or zero. 0 _ + drag them Tangent to a Curve and Derivative of a Function 3

Graphing the Gradient Function Tangent to a Curve and Derivative of a Function Where the gradient of a curve f(x)is positive, we draw the gradient function above the x axis.

4 Graphing the Gradient Function Tangent to a Curve and Derivative of a 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 activity 4

5 Try this one: Tangent to a Curve and Derivative of a Function online activity derivative grapher 5

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

8 Differentiability The process of finding the gradient of a tangent is called differentiation. The resulting function is called the derivative. 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. Tangent to a Curve and Derivative of a Function This function has a 'gap' at x=a, so is not differentiable at when x=a. 8

9 Discuss the differentiability of these functions: Tangent to a Curve and Derivative of a Function rub and reveal 9

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11 Limits A mathematician must know his/her limits. (Old Chinese Proverb) Consider the two equations and graphs below: 1) y = x+2 2) y = x 2 4 =(x 2)(x+2) = x+2 x 2 (x 2) Are they the same? Tangent to a Curve and Derivative of a Function rub and reveal 11

12 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 happens lim x 2 4 = x 2 x = 0 very bad!!! 0 2) Instead, factorise, simplify and THEN substitute lim x 2 4 = lim (x 2)(x+2) x 2 x 2 x 2 (x 2) = lim (x+2) = 4 x 2 Tangent to a Curve and Derivative of a Function 12

13 Important Definition: 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. 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 use the notations lim f(x), and lim f(c + h) x c h 0 to mean the limit of the function as x c. Tangent to a Curve and Derivative of a Function If f(x) is continuous at x = c, then lim f(x) = f(c), and x c lim f(c + h) = f(c), for negative and positive values of h. h 0 13

A limit is used when we want to move as close as we can to something.

14 Using Limits 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. Tangent to a Curve and Derivative of a Function rub and reveal 14

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16 Differentiating by First Principles 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. Tangent to a Curve and Derivative of a Function geogebra activity 16

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18 General Principles 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) x c The gradient of the tangent at P is found when x approaches c. We call this f'(c). Tangent to a Curve and Derivative of a Function 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 = y 2 y 1 x 2 x 1 = f(x+h) f(x) x+h x = f(x+h) f(x) h continued next slide 18

19 As h approaches 0, the gradient of the tangent becomes lim f(x+h) f(x) We call this f'(x) h 0 h 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: Tangent to a Curve and Derivative of a Function 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. 19

20 Differentiate from first principles to find the gradient of the tangent to the curve y = x 2 +3 at the point where x = 1. let y = f(x) f'(x) = lim f(x+h) f(x) h 0 h f(x) = x f(x+h) = (x+h) +3 f'(x) = lim (x+h) +3 (x +3) h 0 h = lim (x 2+2hx+h 2)+3 x2 3 h 0 h = lim (2hx+h ) h 0 h = lim h(2x+h) h 0 h = lim (2x+h) Now substitute in x=1 f'(1) =2 h Tangent to a Curve and Derivative of a Function 20

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22 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 Tangent to a Curve and Derivative of a Function 22

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32 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) = x 5 f'(x) = 5x 4 f(x) = x 4 f'(x) = 4x 3 f(x) = x 3 f'(x) = 3x 2 f(x) = 10x 3 f'(x) = 30x 2 f(x) = 5x 8 f'(x) = 40x 7 brainstorm Rule: Tangent to a Curve and Derivative of a Function 32

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35 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. Tangent to a Curve and Derivative of a Function 35

36 Example of product rule rub and reveal Tangent to a Curve and Derivative of a Function ALWAYS ensure you label 'u' and 'v' before you do anything else! 36

37 Differentiating the quotient of two functions y = u gives the v result: This can also be written as: Quotient Rule Tangent to a Curve and Derivative of a Function 37

38 Example of using quotient rule: rub and reveal Tangent to a Curve and Derivative of a Function 38

Composite Function Rule When you have a function

39 Composite Function Rule When 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: Tangent to a Curve and Derivative of a Function rub and reveal 39

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for differentiation correctly Ensure you can find the equations of tangents and normals to curves at a

45 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 Tangent to a Curve and Derivative of a Function watch a movie Calculus Rhapsody 45

46 Attachments Curves 1st and 2nd Derivative IWB.ggb gradient demo.ggb

Introduction to Calculus

8 Introduction to Calculus TERMINOLOGY Composite function: A function of a function. One function, f (), is a composite of one function to another function, for eample g() Continuity: Describing a line