U4: Contextual Applications of Differentiation

Particle Motion

● s(t) represents the position function, aka f(x)

○ t stands for time, s(t) is the position at a specific time.

● v(t) represents the velocity function, aka f’(x)

○ t stands for time, v(t) is the speed and direction at a specific time.

○ Velocity is the derivative of position.

■ A particle is moving to the right or up when velocity is positive.

■ A particle is moving to the left or down when velocity is negative.

■ A particle’s position is increasing when velocity is positive.

■ A particle’s position is decreasing when its velocity is negative.

■ A particle is at rest or stopped when its velocity is zero

○ Speed is the absolute value of the velocity

● a(t) represents the acceleration function aka f’’(x)

○ t stands for time, a(t) is the rate at which the velocity is changed at specific times.

● Example: s(t)=6t3 - 4t2 → v(t) = 18t2 - 8t → a(t) = 36t - 8

Particle Moving Away/Toward the Origin(x-axis)

● A particle is moving towards the origin when its position and velocity have opposite signs.

● A particle is moving away from the origin when its position and velocity have the same signs,

Particle Speeding Up/Slowing Down

● A particle is speeding up (speed is increasing) if the velocity and acceleration have the same signs at the point.

● A particle is slowing down (speed is decreasing) if the velocity and acceleration have opposite signs at the point.

Related Rates

What is the purpose of related rates?

● The purpose is to find the rate where a quantity changes

● The rate of change is usually with respect to time

How to solve it?

1. Identify all given quantities to be determined.

2. Make a sketch of the situation and label everything in terms of variables, even if you are given actual values.

3. Find an equation that ties your variables together.

4. Using chain rule, implicitly differentiate both sides of the equation with respect to time.

5. Substitute or plug in the given values and solve for the value that is being asked for.

a. *Don’t forget to put the correct units!

The Different Types of Related Rates Problems

● Algebraic

● Circle

● Triangles

● Cube

● Right Cylinder

● Sphere

● Circumference

Sphere:

V = 4π3

SA = 4πr2

Triangles:

a2 + b2 = c2

A = ½ bh

Circles:

A = πr

C = 2πr

Cylinder:

V = πr2h

LSA = 2πrh

SA = 2πrh + 2πr2

Circumference: πd

Cube:

V = s3

SA = 6s2

Cone: V = 3πr2h

Related Rates: Algebraic

Example: A point moves along the curve y = 2x2 - 1 in which y decreases at the rate of 2 units per second. What rate is x changing when x = -3/2?

Related Rates: Circle

Example: The radius of a circle is increasing at a rate of 3cm/sec. How fast is the circumference of the circle changing?

Related Rates: Triangle

Example: A 13ft ladder is leaning against a wall. If the top of the ladder slips down the wall at a rate of 2 ft/s, how fast will the ladder be moving away from the wall when the top is 5ft above the ground?

Related Rates: Cube

Example: The volume of a cube is increasing at a rate of (10cm^3)/min. How fast is the surface area increasing when the length of an edge is 30cm?

Related Rates: Right Cylinder

Example: The radius of a right circular cylinder increases at the rate of 0.1cm/min and the height decreases at the rate of 0.2 cm/mm. What is the rate of change of the volume of the cylinder, in cm^3/min, when the radius is 2cm and the height is 3cm?

Related Rates: Sphere

Example: As a balloon in the shape of a sphere is being blown up, the volume is increasing at a rate of 4in^3/s. At what rate is the radius increasing when the radius is 1 inch.

Related Rates: Circumference

Example: What is the value of the circumference of a circle at the instant when the radius is increasing at 1/6 the rate the area is increasing?

L’Hopital’s Rule

Let f and g be continuous and differentiable functions on an open interval (a,b). If the limit of f(x) and g(x) as x approaches c produces the indeterminate form 0/0 or ∞/∞ then,

lim f(x) = lim f’(x)

x→c g(x) x→c g’(x)

Example: