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Introduction to…. Derivatives Differentiation is one of the most fundamental and powerful operations in all of calculus.

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Presentation on theme: "Introduction to…. Derivatives Differentiation is one of the most fundamental and powerful operations in all of calculus."— Presentation transcript:

1

2 Introduction to…. Derivatives

3 Differentiation is one of the most fundamental and powerful operations in all of calculus.

4 It is a concept that was developed over two hundred years ago by two men… Sir Issac Netwon (Lagrange Notation) Gottfried Leibniz (Leibniz Notation)

5 The output of this operation is called the “derivative”. The derivative can be used to calculate the slope of the tangent at any point in the function’s domain.

6 The formula for the derivative is created through the combination of the 2 main concepts we have studies so far 1. The difference quotient 2. Limits

7 Starting with our difference quotient, we no longer want to construct a single secant line starting at “a”. We want to construct a secant line anywhere within the domain…. To do this, we replace “a” with “x”

8 Starting with our difference quotient, we no longer want to construct a single secant line starting at “a”. We want to construct a secant line anywhere within the domain…. To do this, we replace “a” with “x”

9 Next, we no longer want this to represent a secant line. We want a tangent line. We want to know the exact slope at each point “x”. To do this, we must make h infinitely small. A limit will allow us to reduce h in this manner… m tan = h0 f’(x)= Legrange Notation

10 Video of the DerivativeDerivative

11 Find the derivative of f(x) = x 2 (find a function that represents the slopes of all tangents ) f’(x) = lim h 0 lim h 0 lim h 0 lim h 0 = = =

12 lim h 0 = Since h is not zero, we are free to divide 1 1

13 lim h 0 = 2x We can now sub in any value of x to determine the slope of the tangent for every point x in the domain….. …nice Now “take the limit” substitute h = 0 (always try sub First) = Confirm with sketchpad!!!!!

14 Other notation… Leibniz notation Read as “dee y by dee x” It reminds us of the process by which the derivative is obtained D as in delta, as in “the change in y with respect to the change in x”…

15 Use first principles to differentiate f(x) = x 3 f’(x) = lim h 0

16 The height of a javelin tossed in the air is modelled by the function H(t) = -4.9t 2 + 10t + 1, where H is the height, and t is time, in seconds. a)Determine the rate of change of the height of the javelin at time t. b)Determine the rate of change of the javelin after 1,2 and 3 seconds.

17 Note: A function is also non-differentiable at points where the function has an abrupt change, which is represented by a cusp or a corner on the graph.


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