Logarithmic Differentiation (Sec. 3.6)
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1 Logarithmic Differentiation (Sec. 3.6)
2 Logarithmic Differentiation Use logarithmic differentiation if you are taking the derivative of a function whose formula has a lot of MULTIPLICATION, DIVISION, and/or POWERS in it (or if the problem asks you to use logarithmic differentiation)
3 Logarithmic Differentiation Steps in Logarithmic Differentiation 1) Take the natural logarithm of both sides of the equations 2) Simplify using properties of logs 3) Differentiate both sides with respect to x (implicitly) 4) Solve for y 5) Replace the y in your answer with the original formula for y
4 Logarithmic Differentiation Properties of Logs 1) log a xy = log a x + log a y 2) log a x y = log a x log a y 3) log a x r = r log a x
5 Logarithmic Differentiation Ex 1: Use logarithmic differentiation to find y if a) y = x sinx b) y = x 1 x 4 +1 c) y = xe x2 x (x + 1) 2/3
6 Section 2.1, 2.7, and 3.7: Position Functions, Velocity and Acceleration
7 Position Functions
8 Position Functions Story: An object (like a car) can only travel in a straight line (like a long and narrow road). Imagine placing a number line along the object s path where the 0 on the number line is some reference point (like a tree on the side of the road). The object s position is its location on the number line (where is the car?). Since the object is moving, its position changes with time and so position is a function of time. For a position function, the input is time t (usually in seconds), and the output is position s (usually in feet or meters).
9 What is the position? a) What is the position of the car (assume position is in meters)? Answer: 0 m b) What is the position of the car in words? Answer: The car is at the tree c) If this occurred at time 1 s, what is the notation for the car s position? Answer: s(1)
10 What is the position? a) What is the position of the car (assume position is in meters)? Answer: 1 m b) What is the position of the car in words? Answer: The car is 1 meter to the right of the tree c) If this occurred at time 3 s, what is the notation for the car s position? Answer: s(3)
11 What is the position? a) What is the position of the car (assume position is in feet)? Answer: 4 ft b) What is the position of the car in words? Answer: The car is 4 feet to the right of the tree c) If this occurred at time 7 s, what is the notation for the car s position? Answer: s(7)
12 What is the position? a) What is the position of the car (assume position is in feet)? Answer: -3 ft b) What is the position of the car in words? Answer: The car is 3 feet to the left of the tree c) If this occurred at time 10 s, what is the notation for the car s position? Answer: s(10)
13 Displacement Displacement can only be calculated over a trip (a time interval) Displacement is how far the object is at the end of the trip from where it started at the start of the trip Displacement, unlike a distance, can be negative. Displacement is positive if by the end of the trip the object progressed towards the positive direction compared to where it started Displacement is negative if by the end of the trip the object progressed towards the negative direction compared to where it started
14 What is the displacement? a) What is the displacement of the car over the time interval [1s, 3s]? (assume position is in meters) Answer: 3 m b) What is the notation/formula for the car s displacement over the time interval [1s, 3s]? Answer: s(3) - s(1)
15 What is the displacement? a) What is the displacement of the car over the time interval [1s, 3s]? (assume position is in meters) Answer: -3 m b) What is the notation/formula for the car s displacement over the time interval [1s, 3s]? Answer: s(3) - s(1)
16 What is the displacement? a) What is the displacement of the car over the time interval [3s, 12s]? (assume position is in feet) Answer: 7 ft b) What is the notation/formula for the car s displacement over the time interval [3s, 12s]? Answer: s(12) - s(3)
17 What is the displacement? a) What is the displacement of the car over the time interval [2s, 7s]? (assume position is in feet) Answer: -2 ft b) What is the notation/formula for the car s displacement over the time interval [2s, 7s]? Answer: s(7) - s(2)
18 Formula for displacement Over the trip from time t = a to time t = b, or over the time interval [a,b], the displacement of the object is Displacement = s(b) - s(a)
19 Average Velocity Velocity is almost the same as speed, except it has a direction. Ex: speed = 45 mph velocity = 45 mph East For motion along a line, the direction is indicated by the sign of the answer. Ex: velocity = + 45 mph means 45 mph East, and velocity = - 45 mph means 45 mph West What does average mean? (Story )
20 Average Velocity Average velocity can only be calculated over a trip (a time interval) Average velocity is the constant velocity that the car would have if it traveled straight from its starting position to its ending position with a constant velocity (even though it probably doesn t have a constant velocity)
21 Formula for Average Velocity Over the trip from time t = a to time t = b, or over the time interval [a,b], the average velocity of the object is v ave = s b s(a) b a
22 What is the Average Velocity? What is the average velocity of the car over the time interval [1s, 3s]? (assume position is in meters) Answer: v ave = s 3 s(1) 3 1 = = 1.5 m/s This means that the car s average velocity is 1.5 m/s to the right.
23 What is the Average Velocity? What is the average velocity of the car over the time interval [1s, 3s]? (assume position is in meters) Answer: v ave = s 3 s(1) 3 1 = = 1.5 m/s This means that the car s average velocity is 1.5 m/s to the left.
24 What is the Average Velocity? What is the average velocity of the car over the time interval [3s, 12s]? (assume position is in feet) Answer: v ave = s 12 s(3) 12 3 = 5 ( 2) ft/s This means that the car s average velocity is 0.78 ft/s to the right.
25 What is the Average Velocity? What is the average velocity of the car over the time interval [2s, 7s]? (assume position is in feet) Answer: v ave = s 7 s(2) 7 2 = 3 ( 1) 7 2 = 0.4 ft/s This means that the car s average velocity is 0.4 ft/s to the left.
26 Instantaneous Velocity Imagine an average velocity calculated over a very short time interval (like [3 s, 3.01s]), what would that give you? Instantaneous velocity! Instantaneous velocity formula. v instantaneous = lim t a s t s(a) t a So the derivative of a position function is the instantaneous velocity (or just velocity) function.
27 Ex 1: A ladybug moves in a straight line that runs West-East (East is positive position) with position s t = t 3 11t t for t 0 relative to a rock on the side of the path where t is measured in seconds and s is measured in feet. a) What is the ladybug s position at time t =3s? b) When is the ladybug at the rock? c) When is the ladybug East of the rock? d) When is the ladybug West of the rock? e) What is the displacement of the ladybug over the time interval [2s, 5s]?
28 Ex 1: A ladybug moves in a straight line that runs West-East (East is positive position) with position s t = t 3 11t t for t 0 relative to a rock on the side of the path where t is measured in seconds and s is measured in feet. f) What is the average velocity of the ladybug over the time interval [2s, 5s]? g) What is ladybug s instantaneous velocity function? h) What is the ladybug s instantaneous velocity at time t = 2s? i) When is the ladybug at rest? j) When is the ladybug heading West? k) When is the ladybug heading East?
29 Ex 1: A ladybug moves in a straight line that runs West-East (East is positive position) with position s t = t 3 11t t for t 0 relative to a rock on the side of the path where t is measured in seconds and s is measured in feet. l) What is the distance traveled by the ladybug during the time interval [2s, 10s]? m) What is the average acceleration of the ladybug over the time interval [2s, 4s]? n) What is ladybug s instantaneous acceleration function? o) What is the ladybug s instantaneous acceleration at time t = 2s?
30 Ex 1: A ladybug moves in a straight line that runs West-East (East is positive position) with position s t = t 3 11t t for t 0 relative to a rock on the side of the path where t is measured in seconds and s is measured in feet. p) When is the ladybug s acceleration 0, positive, negative? q) When is the ladybug speeding up? r) When is the ladybug slowing down?
31 Ex 2: If a ball is thrown vertically upward with a velocity of 80 ft/s, then its height after t seconds is s t = 80t 16t 2 for t 0 relative to the floor and positive position is above the ground. a) What is the maximum height reached by the ball? b) What is the velocity of the ball when it is 96 ft above the ground on its way up? On its way down? c) What is the velocity of the ball at the moment right before it hits the ground? d) What is the acceleration of the ball at time t?
32 Ex 3:
33 Ex 4:
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