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Ch 3.2 Differentiability Graphical, Numerical, Algebraic by Finney Demana, Waits, Kennedy Estimate the Derivative Using the function y = ln x, make a table of values for x that would be useful to estimate the derivative of y = ln x at x = 1 and then estimate the derivative. x ln x .97 .98 .99 1 1.01 1.02 1.03 Estimate the Derivative Using the function y = ln x, make a table of values for x that would be useful to estimate the derivative of y = ln x at x = 1 and then estimate the derivative. x .97 .98 .99 1 1.01 1.02 1.03 ln x -.0305 -.0202 -.0101 0 .00995 .0198 .02956 .00995 - (-.0101) .02005 y(1) = = 1.0025 1.01 - .99 .02 1 Local Linearity Activity 1986 AB 4 Let f be the function defined as follows: x - 1 + 2, for x < 1 f(x) = 2 ax + bx, for x 1, where a and b are constants 1. If a = 2 and b = 3, is f continuous for all x? Justify your answer. 2. Describe all values of a and b for which f is a continuous function. 3. For what values of a and b is f both continuous and differentiable? 4. Using these values of a and b and the definition of the derivative, find f '(x) at x = 1.