Higher Maths 1.3

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  • Higher Mathematics:

    Unit 1.3 Introduction to Differentiation

  • Required skillsBefore we start .

    You will need to remember work with Indices, as well as what you have learned about Straight Lines from Unit 1.1.

    Lets recall the rules on indices ..

  • Rules of indices

    1

    RuleExamplesa0 = 112.3140 =a-m = x -5 = =am an = am+n2a3/2 3a1/2= =am an = am-nx2 x -3= =(am)n = amn(q2)3= =

  • What is Differentiation? Differentiation is the process of deriving f (x) from f(x). We will look at this process in a second. f (x) is called the derived function or derivative of f(x). The derived function represents:the rate of change of the functionthe gradient of the tangent to the graph of the function.

  • Tangents to curvesThe derivative function is a measure of the gradient or slope of a function at any given point. This requires us to consider the gradient of a line.

    We can do this if we think about how we measure the gradient from Unit 1.1y2 y1x2 x1The gradient of AB = mAB

    *

  • Tangents to curvesWe will investigate the tangent to a curve at a single point.

    The points A and B lie on a function. The line joining A and B is not a tangent to the curve at the point A. If we think about moving the point B towards A, the slope of the line will change. We can observe the results by watching this graphically. We will take the simple function f(x) = x2

    Click the graphic to investigate this function.

  • Tangents to curvesObserve the table of results and suggest a relationship between the value of the slope of the tangent and the value of x on the function f(x) = x2

    It appears that the relationship may be f (x) = 2x

    Can we find this algebraically?

    xf (x)241200-1-2-2-4

  • Tangents to curvesWe will look at a function and think about the gradient of the function at any given point. The function itself is not important, the process we go through to get the gradient is. We want to find the gradient at a point on a curve.A is the point (x, f(x)) and Bis a point on the function a short distance h from A.This gives B the coordinates(x+h, f(x+h))The line AB is shown on the diagram. We want to find the gradient of the curve at A. If we find the gradient of the line AB and move B towards A we should get the gradient at A.

    xf(x)

  • Tangents to curvesf(x+h)f(x)xx+hThe gradient of the line AB would therefore be

  • Tangents to curvesLook at what happens as we move B towards A, i.e. h is getting smaller.

    As the size of h gets smaller and smaller the line AB becomes the tangent to the curve at the point A. We refer to this as the limit as h 0. Note that h cannot become 0 or we would not get a line AB!

    The gradient of the line AB becomes the same as the gradient of the curve at A

    This is the value we were looking for.

  • Putting it together.Differentiate the functionf(x) = x2 from first principles.

    The derivative is the same as the gradient of the tangent to the curve so we can go straight to the gradient formula we saw in the previous slides.

    The limit as h 0 is written as h gets so small its effectively zero.

  • What does the answer mean?For each and every point on the curve f(x)=x2 , the gradient of the tangent to the curve at the point x is given by the formula f (x)=2xA table of results might make this clearerIs there an easier way to do this?

    -6-4-202468We can agree the observation graphically and algebraically.

    Value of xGradient of the tangent-3-2-101234

  • Spot the pattern2x3x24x35x4nxn-14x6x8x10x2ax9x220x321x242x6anxn-1

    f(x)f (x)f(x)f (x)f(x)f (x)x22x23x3x33x25x4x44x27x3x55x26x7xnax2axn

    f(x)f (x)f(x)f (x)f(x)f (x)x22x23x3x33x25x4x44x27x3x55x26x7xnax2axn

    f(x)f (x)f(x)f (x)f(x)f (x)x22x23x3x33x25x4x44x27x3x55x26x7xnax2axn

  • Rules for differentiationThere are four rules for differentiating remember these and you can differentiate anything 6 x 6 - 1= 6 x 54 x 2 x 2-1= 8 x 1 or 8 x06 x 5 + 8x

    RuleExamplesf(x) = xn f (x) = nxn-1f(x) = x6 f (x) =f(x) = cxn f (x) = cnxn-1f(x)= 4x2 f (x) =

    f(x) = c f (x) = 0f(x) = 65 f (x) = f(x) = g(x) + h(x) f (x) = g (x) + h (x)f(x)= x6 + 4x2 + 65 f (x) =

  • NotationThere are many different ways of writing f (x):

    f (x) y y (x)

    The most common of these are:

    f (x) functional notation used when function is defined as f(x)

    Leibnitz notation used when function is defined as y =

  • The Gradient of a Tangent

    Remember:Differentiation is used to find the gradient of a tangent to a graph. Find the equation of the tangent to the curve y = 3x3 x + 6 at x = 2.A tangent is a straight line, so we need to use:

    y b = m(x a)

    To use this we need to know:

    the gradient of the tangent at the point on the curve

    (in this case when x = 2)

    the coordinates of a point on the line (in this case (2, ?) )

  • The Gradient of a TangentStep 1 :Finding the gradient when x = 2Differentiate the functionAs the function is given in terms of y we will use the dy/dx notationAt the point x = 2, So the gradient of the tangent is 35

  • The Gradient of a TangentStep 2 :Finding the coordinates at the pointAt the point x = 2 the function value is given byy = 3 (2)3 2 + 6

    y = 3(8) + 4

    y = 28The coordinate is therefore the point (2, 28)

  • The Gradient of a TangentStep 3 :Finding the equation of the tangentThe line passing through (2,28) with gradient 35 is :The equation of the tangent at x = 2 is y 35x + 42 = 0

  • The Gradient of a TangentYou can confirm this by checking on a graphing calculator or by sketch.

  • Sketching Graphs of Derived FunctionsWe can investigate graphs of derived functions by looking at a dynamic setting of a function, along with the resulting derived function.

    There are 3 examples shown on separate pages. Look at each one and build up the derived graph as you are prompted. Try to predict the derivative of the third graph before it is drawn.

    Click the graphic to start your investigation

  • Sketching Graphs of Derived FunctionsWe can sketch the graph of the derived function, f (x), by considering the graph of f(x).

    The diagram shows the graph of y = f(x)Sketch the graph of y = f (x)

    Step 1Decide what type of function it is:

    This looks like a cubic function,i.e.it has an x3 term in it.

    When we differentiate the functionthe x3 term will become an x2 term.

    f (x) will be a quadratic function and will look like this or

  • Sketching Graphs of Derived FunctionsStep 2Consider the gradient of the function at key pointsTangent is horizontalm = 0Gradient is positivem > 0Tangent is horizontalm = 0Gradient is positivem > 0Gradient is negativem < 0This can be summarised on a table

  • Sketching Graphs of Derived Functions+ ve0- ve0+ ve

    x-13f (x)

  • Sketching Graphs of Derived FunctionsStep 3Sketch this information as a graphf(x) +ve above the linef(x) -ve below the linef(x) = 0 on the line-13

    x-13f (x)+ ve0- ve0+ ve

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