![• Step 3: Determine the direction of the graph. Determine if there is
relative maximum/minimum.
• Let the function f be continuous on the closed interval [a, b] and
differentiable on the interval (a, b).
• (a) if f’(x) > 0 for all x in (a, b), then f is increasing on [a, b]
• (b) if f’(x) < 0 for all x in (a, b), then f is decreasing on [a, b]
First Derivative Test
(1) The function y = f(x) is a maximum for x = c if
f’(x) > 0 for x < c
f’(x) = 0 for x = c
f’(x) < 0 for x > c
(2) The function y = f(x) is a minimum for x = c if
f’(x) < 0 for x < c
f’(x) = 0 for x = c
f’(x) > 0 for x > c
Sketching of the Curve](https://image.slidesharecdn.com/curve-sketching-160619134828/85/Curve-sketching-3-320.jpg)
Curve sketching
- 2. Sketching of the Curve • Step 1: Determine the domain of f . • Step 2: Find the critical points. Definition: We say that x=c is a critical point of the function f(x) if f(c) exists and if either of the following are true. 𝑓′ 𝑐 = 0 𝑜𝑟 𝑓′ 𝑐 𝑑𝑜𝑒𝑠𝑛′ 𝑡 𝑒𝑥𝑖𝑠𝑡
- 3. • Step 3: Determine the direction of the graph. Determine if there is relative maximum/minimum. • Let the function f be continuous on the closed interval [a, b] and differentiable on the interval (a, b). • (a) if f’(x) > 0 for all x in (a, b), then f is increasing on [a, b] • (b) if f’(x) < 0 for all x in (a, b), then f is decreasing on [a, b] First Derivative Test (1) The function y = f(x) is a maximum for x = c if f’(x) > 0 for x < c f’(x) = 0 for x = c f’(x) < 0 for x > c (2) The function y = f(x) is a minimum for x = c if f’(x) < 0 for x < c f’(x) = 0 for x = c f’(x) > 0 for x > c Sketching of the Curve
- 4. • Step 4: Use the second derivative to find the concavity and the points of inflection. The second derivative is the rate of change of the first derivative. It follows that when y’’ > 0, y’ is increasing. When y’’ < 0, y’ decreases. • The graph of y = f(x) is concave upward on any interval where y’’ > 0 • and concave downward on any interval where y’’ < 0. SECOND DERIVATIVE TEST (1) The function y = f(x) is a maximum at x = c if f’(c) = 0 and f’’(c) < 0. (2) The function y = f(x) is a minimum at x = c if f’(c) = 0 and f’’(c) > 0. Note: if f’’(c) = o or does not exist, then the test fails and we may use the first derivative test. Sketching of the Curve
- 5. • A point where the sense of concavity changes is called a point of inflection. • It can be shown that if y = f(x) has a point of inflection at x = c, then f’’(c) = 0 or f’’(c) does not exist. • POINT OF INFLECTION TEST (1) If f’’(c) = 0 and if f’’(c) > / < for x < c and f’’(c) < / > for x > c, then y = f(x) has a point of inflection at x = c. (2) If f’’(c) = 0 and if f’’(c) 0, then y = f(x) has a point of inflection at x = c. Sketching of the Curve
- 6. • Step 5: Find the horizontal and vertical asymptotes • HA: Get the limit as x approaches to infinity • VA: equate the denominator to zero after dividing out any common factors to find “a” • SA: For rational expressions, slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator. — can be found by division — use y = mx + b and ignore the remainder. Sketching of the Curve
- 7. • Step 6: Find the exact coordinates of the x - and y - intercepts, the critical points, and the points of inflection. • Step 7: Combine all of the evidence collected to graph the function. Sketch the graph Sketching of the Curve
- 8. • Step 1: Determine the domain of f . • Step 2: Find the critical points. • Step 3: Determine the direction of the graph. Determine if there is relative maximum/minimum. • Step 4: Use the second derivative to find the concavity and the points of inflection. • Step 5: Find the asymptotes • Step 6: Find the exact coordinates of the x - and y - intercepts, the critical points, and the points of inflection. • Step 7: Combine all of the evidence collected to graph the function. Sketch the graph Sketching of the Curve