Unit 5 Review - Analytical Applications of Differentiation PDF

Unit 5 Review – Analytical Applications of Differentiation This review summarizes everything from Unit 5 along with examples but contains no problems to work through. DEFINITIONS Extrema: The maximum and minimum points. Extrema can be absolute or relative. Critical Points: Where the first derivat...

Unit 5 Review – Analytical Applications of Differentiation This review summarizes everything from Unit 5 along with examples but contains no problems to work through. DEFINITIONS Extrema: The maximum and minimum points. Extrema can be absolute or relative. Critical Points: Where the first derivative is zero or DNE. These are possible maximum, minimum, or points of inflection! 𝑓 𝑥 0 𝑓 𝑥 0 𝑓 𝑥 0 𝑓 𝑥 𝐷𝑁𝐸 𝑓 𝑥 𝐷𝑁𝐸 Horizontal Horizontal Horizontal Vertical Tangent Cusp (No Tangent Tangent Tangent Point of Inflection Tangent) Maximum Minimum Point of Inflection Maximum Concavity: Where the function is “cupping” up or down CONCAVE UP CONCAVE DOWN Points of Inflection: Where the second derivative is zero or DNE and changes sign! FIRST DERIVATIVE The first derivative is the instantaneous rate of change, or the slope of the tangent line, and can determine if the function is increasing or decreasing at a given point. 𝑓 𝑥 0 𝑓 𝑥 0 𝑓 𝑥 0 Function is increasing Function is not increasing Function is decreasing or decreasing SECOND DERIVATIVE The second derivative determines concavity. 𝑓 𝑥 0 𝑓′′ 𝑥 0 𝑓′′ 𝑥 0 Concave Up Neither concave up or Concave Down concave down FINDING EXTREMA The First Derivative Test EXAMPLE STEPS 𝒇 𝒙 𝒙𝟐 𝟐𝒙 𝟏 𝑓 𝑥 2𝑥 2 1. Find the critical points. 0 2𝑥 2 𝑥 1 Interval ∞, 𝟏 𝟏 𝟏, ∞ 2. Determine whether the function is Test Value 2 1 2 increasing or decreasing on each side of 𝑓 2 𝑓 1 𝑓 2 6 every critical point. 𝑓′ 𝑥 Negative 0 Positive A chart or number line helps! Function decreases to the left and increases to the right of 𝑥 1 so it must be relative minimum point The Second Derivative Test EXAMPLE STEPS 𝒇 𝒙 𝒙𝟐 𝟐𝒙 𝟏 𝑓 𝑥 2𝑥 2 1. Find the critical points. 0 2𝑥 2 𝑥 1 2. Determine whether the function is concave up 𝑓′′ 1 2 or concave down at every critical point using Second derivative is positive at 𝑥 1 the second derivative. Concave up 𝑥 1 is a relative minimum point Finding Absolute Extrema on an Interval (Candidates Test) EXAMPLE STEPS 𝒇 𝒙 𝒙𝟐 𝟐𝒙 𝟏 on the interval 𝟑, 𝟎 1. Find the critical points. The critical points are 𝑓 𝑥 2𝑥 2 candidates as well as the endpoints of the 0 2𝑥 2 interval. 𝑥 1 𝑓 3 4 absolute maximum 2. Check all candidates using the 𝑓 𝑥. 𝑓 1 0 absolute minimum 𝑓 0 1 LINEAR MOTION The chart matches up function vocabulary with linear motion vocabulary. FUNCTION LINEAR MOTION Value of a function at x Position at time t First Derivative Velocity Second Derivative Acceleration 𝑓 𝑥 0 Moving right or up Increasing Function 𝑓 𝑥 0 Moving left or down Decreasing Function 𝑓 𝑥 0 Not moving Absolute Max Farthest right or up Absolute Min Farthest left or down 𝑓 𝑥 changes signs Object changes direction 𝑓 𝑥 and 𝑓 𝑥 have same Speeding Up sign 𝑓 𝑥 and 𝑓 𝑥 have Slowing Down different signs Example: A particle moves along the x-axis with the position function 𝑥 𝑡 𝑡 4𝑡 2 where 𝑡 0. Interval 𝟎, 𝟐 𝟐 𝟐, 𝟑 𝟑 𝟑, ∞ 𝑥 𝑡 0 𝑥 𝑡 0 𝑥 𝑡 0 𝑥 𝑡 0 𝒙 𝒕 𝑥 𝑡 0 increasing Increasing increasing decreasing velocity Not moving right right right left 𝑥 𝑡 0 𝑥 𝑡 0 𝑥 𝑡 0 𝒙 𝒙 𝑥 𝑡 0 𝑥 𝑡 0 Concave Concave Concave acceleration Concave up down down down Speeding Moving Slowing Not Speeding Conclusion Up Right Down Moving Up FUNCTION LINEAR MOTION 𝑡 3 has no velocity 𝑡 3 is maximum Changing direction Increasing 0,3 Moving right 0,3 Decreasing 3, ∞ Moving left 3, ∞ GRAPHICAL ANALYSIS Connecting 𝒇 𝒙 to 𝒇 𝒙 𝒇 𝒙 𝒇′ 𝒙 𝑓 𝑥 max at 𝑥 2.2 and min 𝑥 2.2 so 𝑓 𝑥 0 at these points 𝑓 𝑥 is increasing on ∞, 2.2 2.2, ∞ so 𝑓 𝑥 0 on these intervals 𝑓 𝑥 is decreasing on 2.2, 2.2 so 𝑓 𝑥 0 on this interval Connecting 𝒇 𝒙 to 𝒇 𝒙 𝒇 𝒙 𝒇 𝒙 𝑓 𝑥 is concave down on ∞, 0 so 𝑓′′ 𝑥 0 on this interval 𝑓 𝑥 is concave up on 0, ∞ so 𝑓 𝑥 0 on this interval 𝑥 0 is a point of inflection on 𝑓 𝑥 so 𝑓′′ 𝑥 changes sign at 𝑥 0. Connecting 𝒇 𝒙 to 𝒇 𝒙 𝒇 𝒙 𝒇 𝒙 𝑓′ 𝑥 min 𝑥 0 so 𝑓′′ 𝑥 0 at 𝑥 0 𝑓′ 𝑥 is decreasing on ∞, 0 so 𝑓′′ 𝑥 0 on this interval 𝑓′ 𝑥 is increasing on 0, ∞ so 𝑓′′ 𝑥 0 on this interval  Using 𝒇 𝒙 to draw conclusions about 𝒇 𝒙 𝒇 𝒙 Find Extrema of 𝒇 𝒙 𝑥 2 and 4 are critical points because 𝑓 𝑥 0 𝑥 2 is a maximum because 𝑓′ 𝑥 is positive on left, negative on right 𝑥 4 is NOT an extrema because 𝑓′ 𝑥 is negative on left, negative on right Find Points of Inflection of 𝒇 𝒙 𝑥 1, 3, and 4 are possible points of inflection 𝑥 3 is a point of inflection because because 𝑓′′ 𝑥 changes sign from negative to positive at 𝑓 𝑥 0 or 𝐷𝑁𝐸 𝑥 3. 𝑥 1 is a point of inflection because 𝑥 4 is a point of inflection because 𝑓′′ 𝑥 changes sign from positive to negative at 𝑓′′ 𝑥 changes sign from positive to negative at 𝑥 1. 𝑥 4. Now interpret the same graph as linear motion if the graph represents velocity of a particle moving along x-axis. 𝒙′ 𝒕 Moving Right or Left? Particle moves right on 0,2. 𝑡 2 particle changes direction. Particle moves left on 2,4 4,7. 𝑡 4 particle has no velocity. The maximum speed happens at 𝑡 3. Speeding up or Slowing down? Particle speeds up on 0,1 because 𝑓 𝑥 has the same sign as 𝑓 𝑥 Particle speeds up on 2,3 because 𝑓 𝑥 has the same sign as 𝑓 𝑥 Particle slows down on 1,2 because 𝑓 𝑥 has a different sign from 𝑓 𝑥 Particle slows down on 3,4 because 𝑓 𝑥 has a different sign from 𝑓 𝑥 Particle speeds up on 4,7 because 𝑓 𝑥 has the same sign as 𝑓 𝑥