Clicker Question 1 What is an equation of the tangent line to the curve f (x ) = x 2 at the point (1, 1)? A. y = 2x B. y = 2 C. y = 2x 2 D. y = 2x + 1.
Published byModified over 6 years ago
Presentation on theme: "Clicker Question 1 What is an equation of the tangent line to the curve f (x ) = x 2 at the point (1, 1)? A. y = 2x B. y = 2 C. y = 2x 2 D. y = 2x + 1."— Presentation transcript:
1 Clicker Question 1What is an equation of the tangent line to the curve f (x ) = x 2 at the point (1, 1)?A. y = 2xB. y = 2C. y = 2x 2D. y = 2x + 1E. y = 2x – 1
2 The Derivative Function (2/13/09) Since we know how to compute (or at least estimate) the derivative (i.e., the instantaneous rate of change) of a given function f at any fixed point, we can then view the derivative itself as a function .We denote this function by f ' (we say “f prime”).Given an input x , it outputs the rate of change at x .
3 The Derivative Function Definition:The derivative function can be :Graphed, given a graph of fEstimated, given numerical info about fComputed algebraically, given a formula for f
4 Derivatives of very simple functions What is the derivative of any constant function f (x) = c at any point x ?What is the derivative of any linear function f (x) = m x + b at any point x ?That is, again, you don’t need calculus to understand the rate of change of linear functions.
5 Derivatives of Some Algebraic Functions Use the definition of the derivative to find the derivative of the function f (x) = x 3Use the definition to find the derivative of f (x) = 1 /x .Use the definition to find the derivative of f (x) = x .
6 Clicker Question 2Using what we just learned, what is the instantaneous rate of change of the function f (x ) = x 3 at the point (2, 8)?A. 3x 2B. 8C. 12D. 12xE. 12x 2
7 DifferentiabilityA function f is said to be differentiable at x = a if f '(a ) exists.How can f not be differentiable at a ??If the graph of f breaks apart at a (we say f is not continuous at a ).If the graph has a sharp bend at a .If the graph becomes vertical at a .
8 Clicker Question 3 The function f (x ) = |x | is not differentiable A. at x = -1B. at x = 0C. at x = 1D. at all three of these pointsE. everywhere
9 AssignmentWe will have Lab #3 on Monday on rates of change of functions.For Wednesday, please read Section 2.8 and do Exercises 1, 3, 5, 9, 14, 16, 19, 21, 25, 27, 29, 33a, 35, and 51.