3.2 Derivative as a Function
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This document covers key concepts in calculus including: - Computing derivatives using notation such as f'(a) and dy/dx - Relationships between differentiability and continuity - Using derivatives to find horizontal tangents, max/min points, and inflection points - Applying derivative rules such as the sum and constant multiple rules - Examples of computing derivatives of various functions and determining differentiability
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3.2 Derivative as a Function
- 1. AP Calculus
- 2. Derivative f’ (a) for specific values of a
- 3. Function f’ (x)
- 4. Domain of f’ (x) is all values of x in domain of f (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.
- 5. Prove that f (x) = x3 – 12x is differentiable. Compute f ‘(x) and write the equation of the tangent line at x = -3.
- 6. F ‘(x) = 3x2 – 12 Equation of tangent line at x = -3 y = 15x + 54
- 7. Calculate the derivative of y = x-2. Find the domain of y and y’
- 8. Solution: y’ = -2x-3 Domain of y: {x| x ≠ 0} Domain of y’ : {x| x ≠ 0} The function is differentiable.
- 9. Another notation for writing the derivative: Read “dy dx” For the last example y = x-2, the solution could have been written this way:
- 10. For all exponents n,
- 12. Calculate the derivative of the function below
- 13. Solution:
- 14. 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’
- 15. Find the points on the graph of f(t) = t3 – 12t + 4 where the tangent line(s) is horizontal.
- 16. Solution:
- 17. How is the graph of f(x) = x3 – 12x related to the graph of f’(x) = 3x2 – 12 ?
- 18. 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
- 19. Differentiability Implies Continuity If f is differentiable at x = c, then f is continuous at x = c.
- 20. Show that f(x) = |x| is continuous but not differentiable at x = 0.
- 21. The function is continuous at x = 0 because
- 22. The one-sided limits are not equal: The function is not differentiable at x = 0
- 23. Local Linearity f(x) = x3 – 12x
- 24. g(x) = |x|
- 25. Show that f(x) = x 1/3 is not differentiable at x = 0.
- 26. The limit at x = 0 is infinite f’(0) = The slope of the tangent line is infinite – vertical tangent line