Assigned work: pg 74#5cd,6ad,7b,8-11,14,15,19,20 Slope of a tangent to any x value for the curve f(x) is: This is know as the “Derivative by First Principles”

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Presentation transcript:

Assigned work: pg 74#5cd,6ad,7b,8-11,14,15,19,20 Slope of a tangent to any x value for the curve f(x) is: This is know as the “Derivative by First Principles” S. Evans

Differentiability Symbols for Derivative: S. Evans

Differentiability Ex 1: Find the equation of the normal to the curve S. Evans Note: An linear equation needs a slope and a point. To get the slope take the derivative. To get a point find f(-2). A normal is a line at a point on f(x) perpendicular to the tangent line at that point. See solution next slide

Differentiability S. Evans

Differentiability S. Evans Graphs of Derivatives: The graph of a derivative is the graph of the changing slopes of the tangent. Ex 2: Given the following graphs sketch the graph of the derivative: a)

Differentiability S. Evans b)

When does a derivative exist? For a function, to be Differentiable at x=c: Left hand limit = Right hand limit of BOTH AND the function MUST be continuous at x = c Differentiability Implies Continuity If a function is differentiable at x=c then it MUST be continuous at x=c BUT if a function is continuous it does not have to be differentiable. S. Evans

Differentiability The following are NOT differentiable at x = a a aa CUSP VERTICAL TANGENTDISCONTINUOUS S. Evans

Differentiability Mr.Function knew that so he took the slopes from the s-t graph to get a v-t graph and then the slopes from the v-t graph to get the a-t graph. Mr. Calculus said “there is a problem with Mr. Function’s graphs”. Explain in terms of differentiability what the problem is. S. Evans Position Time Graph s-t Velocity Time Graph v-t Acceleration Time Graph a-t

Differentiability Ex. 1: For the following piecewise function determine if the function is differentiable at a) x=6 b) x= 4 c) x= 2 d) x=0 S. Evans

Differentiability a) Since for x=6.. Therefore, is differentiable at x = 6. S. Evans

Differentiability b) At x = 4.. BUT is NOT continuous at x = 4 Therefore, is NOT differentiable at x = 4. S. Evans

Differentiability c) At x = 2.. Therefore, is NOT differentiable at x = 2. d) At x = 0 Therefore, is NOT differentiable at x = 2. S. Evans