4.
Domain of f’ (x) is all values of x in domain off (x) for which the limit exists. F’ (x) is differentiable on (a, b) if f ‘(x) exists for all x in (a, b). If f’ (x) exists for all x, then f (x) is differentiable.
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Prove that f (x) = x3 – 12x is differentiable. Compute f ‘(x) and write the equation of the tangent line at x = -3.
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F ‘(x) = 3x2 – 12 Equation of tangent line at x = -3 y = 15x + 54
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Calculate the derivative of y = x-2. Find the domain of y and y’
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Solution: y’ = -2x-3 Domain of y: {x| x ≠ 0} Domain of y’ : {x| x ≠ 0} The function is differentiable.
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Another notation for writing the derivative: Read “dy dx” For the last example y = x-2, the solution could have been written this way:
13.
Assume that f and g are differentiable functions. Sum Rule: the function f + g is differentiable (f + g)’ = f’ + g’ Constant Multiple Rule: For any constant c, cf is differentiable and (cf)’ = cf’
14.
Find the points on the graph of f(t) = t3 – 12t + 4 where the tangent line(s) is horizontal.
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How is the graph of f(x) = x3 – 12x related to the graph of f’(x) = 3x2 – 12 ?
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f(x) = x3 – 12 x Decreasing on (-2, 2) Increasing on (2, ∞) Increasing on (-∞, -2) What happens to f(x) at x = -2 f’(x) = 3x2 - 12 and x = 2??Graph of f’(x) positive f’(x) is negative f’(x) is positive on (2, ∞)on (-∞, -2) on (-2,2) Zeros at -2, 2
18.
Differentiability Implies Continuity If f is differentiable at x = c, then f is continuous at x = c.
19.
Show that f(x) = |x| is continuous but not differentiable at x = 0.
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