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Material Taken From: Mathematics for the international student Mathematical Studies SL Mal Coad, Glen Whiffen, John Owen, Robert Haese, Sandra Haese and Mark Bruce Haese and Haese Publications, 2004

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Objectives Describe what the first derivative yields. Find the first derivative of functions. Write equations of the line tangent to a curve.

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Introductory Videos Brainpop.com o Calculus Khan Academy o Calculus (Derivatives 1) Section 19A-F – Introduction to Calculus

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Consider point A (a, f(a)) and point B (x, f(x)) slope of the tangent line at A

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Main Ideas: The derivative of a curve at a point is the slope of the tangent line at that point. The derivative function is the function that represents the slopes of all the tangents throughout the entire curve.

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Graphing Example 1

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Graphing Example 2 What is the function or equation for this graph? What is the derived function of this graph? What would that function look like? What would that function represent?

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name of original functionname of derivative function f(x)f(x)f’(x) yy' y Notation and Terminology Differentiation is the process of finding the derivative or the derivative function (the slope function). Notation

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FunctionDerivative Function f(x) = x f’(x) = f(x) = x 2 f’(x) = f(x) = x 3 f’(x) = f(x) = x 4 f’(x) = f(x) = x 5 f’(x) = f(x) = x -1 f’(x) = f(x) = x -2 f’(x) = f(x) = x -3 f’(x) = f(x) = f’(x) = f(x) = f’(x) =

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Rules (Page 615) f(x)f’(x)in words: a0 The derivative of a constant is zero. xnxn nx n-1 Bring down the power, subtract one from the power. ax n anx n-1 Multiply by the coefficient (same as above). u(x) + v(x)u’(x) + v’(x) The derivative of the sum is the sum of the derivatives.

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Find the first and second derivative of f(x) = 3x 4 + 2x 3 – 5x 2 + 7x + 6 Example 1

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Find the first and second derivative of f(x) = 5x 3 + 6x 2 – 3x+ 2 Example 2

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Find the derivative of Hint: First rewrite the function, then take the derivative. Example 3

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Find the slope function of Hint: First rewrite the function, then take the derivative. Example 4

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a) Find the gradient function of and then find: b) gradient of the tangent to the function where x = 2. c) equation of the tangent when x = 2. Example 5

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a) Find the gradient function of and then find: The gradient function is the first derivative. Now find the gradient when x = 2. Therefore, m = 5 Finally, find the equation of the line. You need a point. We already have the slope. So, the point is (2, 2) and the slope is 5. Therefore, the equation of the tangent at x = 2 is y = 5x – 8. b) gradient of the tangent to the function where x = 2. c) equation of the tangent when x = 2. Example 5 Answer

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Find the equation of the tangent to f(x) = x 2 + 1 at the point where x = 1 Example 6

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Find the equations of any horizontal tangents to y = x 3 – 12x + 2 Example 7

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Summary The rate of change at a point is the slope of the tangent line at that point. The slope of the tangent line at that point is known as the derivative. Therefore: – rate of change = slope of tangent line = derivative

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Homework 19E, #1a-l, 2a-f 19F #1a-d, 2abc, 3abc Kahn Academy Video – Calculus: Derivatives 3

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