Section 3.3 Increasing/Decreasing & The 1 st Derivative Test Day 1 Investigation 1 (Calculator): f(x) = x 2 3x + 4 State all extremes on [0, 5]: Original graph: Global min(s): Global max(s): Local min(s): Local Max(s): Intervals of Increasing: Intervals of Decreasing: Derivative Function: Derivative Graph: Critical Numbers: Investigation 2 (Calculator): f(x) = 2sin(0.5x) Original graph: State all extremes on [-2!, 2!]: Global min(s): Global max(s): Local min(s): Local Max(s): Intervals of Increasing: Intervals of Decreasing: Derivative Function: Derivative Graph: Critical Numbers: Where is the derivative Positive? Negative? Conclusion:
Testing for Increasing If f(x) is the original and f (x) is the derivative, then f(x) will be increasing anywhere that f (x) is positive [when f (x) > 0]. Testing for Decreasing If f(x) is the original and f (x) is the derivative, then f(x) will be decreasing anywhere that f (x) is negative [when f (x) < 0]. Note: to apply these tests on an interval from a to b, the function must be continuous on [a, b] and differentiable on (a, b). Picture: Consider f(x) = (1/3)x 3 + (5/2)x 2 + 6x 1 Example from 3-3A: 3 Your Turn from 3-3A: 4 HW Investigation: The graphs below represent 3 functions and the derivatives of those functions. Identify the original functions and state which is the derivative of each. Use Complete Sentences to describe your reasoning. A. B. C. D. E. F.
Section 3.3 Increasing/Decreasing & The 1 st Derivative Test Day 2 Memorization Practice: First quadrant of unit circle (rad and deg): Product Rule: Chain Rule: 3 parts of continuity at c using limits define critical number what might happen at a critical number? How can you tell if f(x) is increasing? State the Mean Value Theorem First Derivative Test 1. Find all critical numbers. Note: the function must be continuous at these critical numbers 2. Create a number line with the critical points marked. The first derivative test will check if these values are relative min s or max s. 3. Test values between the critical points (choose easy numbers) in the DERIVATIVE and write down if the y value is positive or negative on the number line. 4. Interpret the number line. The ORIGINAL will be increasing when the DERIVATIVE is positive. The ORIGINAL will be decreasing when the DERIVATIVE is negative. If the ORIGINAL changes from increasing to decreasing, then there is a relative maximum at the critical point. If the ORIGINAL changes from decreasing to increasing, then there is a relative minimum at the critical point. Picture:
Examples from 3-3A: 15, 30, Your Turn from 3-3A: 32, Assignment: Finish 3-3A (Front: 3, 4, 44, 46, 15, 30, 32; Back: 16 and 16) Section 3.3 Increasing/Decreasing & The 1 st Derivative Test Day 3 Examples from 3-3A: 15, 30, 49, 44 Your Turn from 3-3A: 32, 50, 46 Assignment: Finish 3-3A Section 3.3 Increasing/Decreasing & The 1 st Derivative Test Day 4 Draw a picture that illustrates the first derivative test: Examples from 3-3B: 14 Your Turn from 3-3B: 88 Assignment: Finish 3-3B
Section 3.4 Concavity & The 2 nd Derivative Test Day 1 Concavity The curvature of the function. Concrete Example: if the graph is a road that you are traveling upon in a car, then the steering wheel will likely be turned left or right. When the steering wheel changes from turning left to turning right (or R to L) the concavity changes. Concave up means it is curving upward (steering wheel turned left). Think of a smiling parabola. Concave down means it is curving downward (steering wheel turned right). Think of a frowning parabola. Pictures: Inflection Point: Where f (x) = 0. Inflection points are potential changes in concavity. Testing for Concavity: If f (x) > 0 then f(x) is concave up. If f (x) < 0 then f(x) is concave down.
Example Consider f(x) = (1/6)x^3 + (3/2)x^2 x + 1 f(x) f (x) f (x) Examples from 3-4A: 1, 5, 12 Examples from 3-4A: 2, 6, 14 Assignment: finish 3-4A (front: stop at 16; back: 5 and 29) Section 3.4 Concavity & The 2 nd Derivative Test Day 2 Examples from 3-4A: 18, 53 Examples from 3-4A: 20, 54 Assignment: finish 3-4A Section 3.4 Concavity & The 2 nd Derivative Test Day 3 Draw a Sine and Cosine Curve from 0 to 2! Power Rule: Quotient Rule: 3 ways a function is not differentiable define critical number when will a function be decreasing? State the first derivative test and draw a picture
Second Derivative Test 1. Find all critical numbers. Note: the function must be continuous at these critical numbers 2. Plug the critical numbers into the second derivative. 3. Interpret the answers. Picture: The ORIGINAL will have a relative minimum when the 2 nd DERIVATIVE is positive because the ORIGINAL is concave up. The ORIGINAL will have a relative maximum when the 2 nd DERIVATIVE is negative because the ORIGINAL is concave down. If the 2 nd DERIVATIVE is zero, then you must use the 1 st derivative test to locate extrema. Examples from 3-4B: 11 Your Turn from 3-4B: 13 Assignment: Finish 3-4B (front: 11-26; back: 2 and 7) Section 3.4 Concavity & The 2 nd Derivative Test Day 4 Examples from 3-4B: 27, 28 Your Turn from 3-4B: 29 Assignment: Finish 3-4B
Section 3.4 Concavity & The 2 nd Derivative Test Day 3 Unit circle quad 1: f(x) = (4x 1)(sin x) f (x) = f(x) = (tan x) 2 / sec (2x) f (x) = How do I find intervals of inc. and dec? Examples from 3-4C: 45, 49, 51 Your Turn from 3-4C: 46, 50, 52 Assignment: Finish 3-4C Section 3.4 Concavity & The 2 nd Derivative Test Day 4 Sin!/6 = sin!/3 = cos!/2 = cos!/4 = Draw a function with domain [-2, 2] that is not continuous at -1 and not differentiable at 1 define critical number define an inflection point How do I find intervals of concavity? Examples from 3-4D: 55, 58 Your Turn from 3-4D: 56, 57 Assignment: Finish 3-4D Study for Quiz 6