Index FAQ The derivative as the slope of the tangent line (at a point)

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Definition of the Derivative Using Average Rate () a a+h f(a) Slope of the line = h f(a+h) Average Rate of Change = f(a+h) – f(a) h f(a+h) – f(a) h.

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Presentation transcript:

Index FAQ The derivative as the slope of the tangent line (at a point)

Index FAQ Video help: MIT!!! sc-single-variable-calculus-fall- 2010/1.-differentiation/

Index FAQ What is a derivative? A function, which gives the: the rate of change of a function in general the slope of the line tangent to the curve in general

Index FAQ What is a differential quotient? Just a number! the rate of change of a function at a given point the slope of the line tangent to the curve at a certain point The substitutional value of the derivative

Index FAQ The tangent line single point of intersection

Index FAQ slope of a secant line a x f(x) f(a) f(a) - f(x) a - x

Index FAQ slope of a (closer) secant line ax f(x) f(a) f(a) - f(x) a - x x

Index FAQ closer and closer… a

Index FAQ watch the slope...

Index FAQ watch what x does... a x

Index FAQ The slope of the secant line gets closer and closer to the slope of the tangent line...

Index FAQ As the values of x get closer and closer to a! a x

Index FAQ The slope of the secant lines gets closer to the slope of the tangent line......as the values of x get closer to a Translates to….

Index FAQ lim ax f(x) - f(a) x - a Equation for the slope Which gives us the the exact slope of the line tangent to the curve at a! as x goes to a

Index FAQ similarly... a a+h f(a+h) f(a) f(x+h) - f(x) (x+h) - x = f(x+h) - f(x) h (For this particular curve, h is a negative value) h