Logarithmic Differentiation (Sec. 3.6) 1 Lecture Notes And Videos/M… · Use logarithmic differentiation if you are taking the ... What is the distance traveled by the ladybug during

Logarithmic

Differentiation

(Sec. 3.6)

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)


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


Properties of Logs

1) log𝑎 𝑥𝑦 = log𝑎 𝑥 + log𝑎 𝑦

2) log𝑎𝑥

𝑦= log𝑎 𝑥 − log𝑎 𝑦

3) log𝑎𝑥𝑟 = 𝑟 log𝑎 𝑥


Ex 1: Use logarithmic differentiation to find 𝑦′ if …

a) 𝑦 = 𝑥𝑠𝑖𝑛𝑥

b) 𝑦 =𝑥−1

𝑥4+1

c) 𝑦 = 𝑥𝑒𝑥2−𝑥(𝑥 + 1)2/3

Section 2.1, 2.7, and 3.7:

Position Functions, Velocity

and Acceleration

Position Functions

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).

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)



in meters)? Answer: 1 m


Answer: The car is 1 meter to the right of the tree





in feet)? Answer: 4 ft


Answer: The car is 4 feet to the right of the tree





in feet)? Answer: -3 ft


Answer: The car is 3 feet to the left of the tree



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

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)




Answer: -3 m



Answer: s(3) - s(1)



interval [3s, 12s]? (assume position is in feet)

Answer: 7 ft



Answer: s(12) - s(3)




Answer: -2 ft



Answer: s(7) - s(2)

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)

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…)

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)

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…

𝑣𝑎𝑣𝑒 = 𝑠 𝑏 −𝑠(𝑎)

𝑏−𝑎

What is the Average Velocity?

What is the average velocity of the car over the time


Answer: 𝑣𝑎𝑣𝑒 =𝑠 3 −𝑠(1)

3−1=

4−1

3−1= 1.5 𝑚/𝑠

This means that the car’s average velocity is 1.5 m/s

to the right.





3−1=

1−4

3−1= −1.5 𝑚/𝑠

This means that the car’s average velocity is 1.5 m/s

to the left.





12−3=

5−(−2)

12−3≈ 0.78 𝑓𝑡/𝑠

This means that the car’s average velocity is 0.78 ft/s

to the right.





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.

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.

𝑣𝑖𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛𝑒𝑜𝑢𝑠 = lim𝑡→𝑎

𝑠 𝑡 − 𝑠(𝑎)

𝑡 − 𝑎

• So the derivative of a position function is the

instantaneous velocity (or just velocity) function.

Ex 1: A ladybug moves in a straight line that runs

West-East (East is positive position) with position

𝑠 𝑡 = 𝑡3 − 11𝑡2 + 24𝑡

relative to a rock on the side of the path where t is

measured in seconds and s is measured in feet.

for 𝑡 ≥ 0

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]?



𝑠 𝑡 = 𝑡3 − 11𝑡2 + 24𝑡



for 𝑡 ≥ 0

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?



𝑠 𝑡 = 𝑡3 − 11𝑡2 + 24𝑡



for 𝑡 ≥ 0

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?



𝑠 𝑡 = 𝑡3 − 11𝑡2 + 24𝑡



for 𝑡 ≥ 0

p) When is the ladybug’s acceleration 0, positive,

negative?

q) When is the ladybug speeding up?

r) When is the ladybug slowing down?

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 − 16𝑡2

relative to the floor and positive position is above the

ground.

for 𝑡 ≥ 0

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 ?

Ex 3:

Ex 4:

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Logarithmic Differentiation (Sec. 3.6) 1 Lecture Notes And Videos/M… · Use logarithmic differentiation if you are taking the ... What is the distance traveled by the ladybug during