Benginning Calculus Lecture notes 9 - derivative functions
- 1. Beginning Calculus - The First and Second Derivative Functions - Shahrizal Shamsuddin Norashiqin Mohd Idrus Department of Mathematics, FSMT - UPSI (LECTURE SLIDES SERIES) VillaRINO DoMath, FSMT-UPSI (D7) The First and Second Derivative Functions 1 / 15
- 2. The First Derivative Functions The Second Derivative Functions Learning Outcomes Interprete the meaning of …rst and second derivative functions. Sketch the graphs of …rst and second derivative functions. VillaRINO DoMath, FSMT-UPSI (D7) The First and Second Derivative Functions 2 / 15
- 3. The First Derivative Functions The Second Derivative Functions The First Derivative Functions If f 0 > 0, then f is increasing. If f 0 < 0, then f is decreasing. VillaRINO DoMath, FSMT-UPSI (D7) The First and Second Derivative Functions 3 / 15
- 4. The First Derivative Functions The Second Derivative Functions Example The graphs of f (x) = x2 and its derivative function f 0 (x) = 2x : -4 -2 2 4 -4 -2 2 4 x y The graph of f -4 -2 2 4 -4 -2 2 4 x y The graphs of f and f 0 VillaRINO DoMath, FSMT-UPSI (D7) The First and Second Derivative Functions 4 / 15
- 5. The First Derivative Functions The Second Derivative Functions Example The graphs of f (x) = x3 3 and its derivative function f 0 (x) = 3x2 1 : -4 -2 2 4 -10 -5 5 10 x y The graph of f -4 -2 2 4 -4 -2 2 4 x y The graphs of f and f 0 VillaRINO DoMath, FSMT-UPSI (D7) The First and Second Derivative Functions 5 / 15
- 6. The First Derivative Functions The Second Derivative Functions Example The graphs of y = 1 x and dy dx = 1 x2 : -4 -2 2 4 -2 -1 1 2 x y The graph of y -4 -2 2 4 -2 -1 1 2 x y The graphs of y and dy dx VillaRINO DoMath, FSMT-UPSI (D7) The First and Second Derivative Functions 6 / 15
- 7. The First Derivative Functions The Second Derivative Functions Example The graphs of f (x) = p x and its derivative function f 0 (x) = 1 2 p x : -4 -2 2 4 -4 -2 2 4 x y The graph of f -4 -2 2 4 -4 -2 2 4 x y The graphs of f and f 0 VillaRINO DoMath, FSMT-UPSI (D7) The First and Second Derivative Functions 7 / 15
- 8. The First Derivative Functions The Second Derivative Functions Example The graphs of f (x) = 1 x 2 + x and its derivative function f 0 (x) = 3 (x + 2)2 : -4 -2 2 4 -4 -2 2 4 x y The graph of f -4 -2 2 4 -4 -2 2 4 x y The graphs of f and f 0 VillaRINO DoMath, FSMT-UPSI (D7) The First and Second Derivative Functions 8 / 15
- 9. The First Derivative Functions The Second Derivative Functions Example Sketch the graph of the derivative function of the given function: -4 -2 2 4 -4 -2 2 4 x y The graph of f -4 -2 2 4 -4 -2 2 4 x y The graphs of f and f 0 VillaRINO DoMath, FSMT-UPSI (D7) The First and Second Derivative Functions 9 / 15
- 10. The First Derivative Functions The Second Derivative Functions Example Sketch the graph of the derivative function of the given function: -4 -2 2 4 -4 -2 2 4 x y The graph of f -4 -2 2 4 -4 -2 2 4 x y The graphs of f and f 0 VillaRINO DoMath, FSMT-UPSI (D7) The First and Second Derivative Functions 10 / 15
- 11. The First Derivative Functions The Second Derivative Functions The Second Derivative Function If f 00 > 0, then f 0 is increasing. The graph of f is concave up. If f 00 < 0, then f 0 is decreasing. The graph of f is concave down. 0>′f 0<′f 0=′f 0<′′f Concave Down Concave Up 0<′f 0>′f 0>′′f 0=′f Inflection Point 0=′′f VillaRINO DoMath, FSMT-UPSI (D7) The First and Second Derivative Functions 11 / 15
- 12. The First Derivative Functions The Second Derivative Functions Example f (x) = x2, Then, f 0 (x) = 2x and f 00 (x) = 2 -4 -2 2 4 -5 5 10 x y f(x) f '(x) f ''(x) VillaRINO DoMath, FSMT-UPSI (D7) The First and Second Derivative Functions 12 / 15
- 13. The First Derivative Functions The Second Derivative Functions Example f (x) = p x, Then, f 0 (x) = 1 2 p x , and f 00 (x) = 1 4 p x 3 2 4 -4 -2 0 2 4 x y f(x) f '(x) f ''(x) VillaRINO DoMath, FSMT-UPSI (D7) The First and Second Derivative Functions 13 / 15
- 14. The First Derivative Functions The Second Derivative Functions Example Sketch the second derivative function: -4 -2 2 4 -4 -2 2 4 x y f(x) f '(x) VillaRINO DoMath, FSMT-UPSI (D7) The First and Second Derivative Functions 14 / 15
- 15. The First Derivative Functions The Second Derivative Functions Example Sketch the second derivative function: -4 -2 2 4 -3 -2 -1 1 2 3 x y f(x) f '(x) VillaRINO DoMath, FSMT-UPSI (D7) The First and Second Derivative Functions 15 / 15