AP Calc Unit 4: Contextual Applications of Derivatives

Describe dy/dx in context

the ROC of the dependent variable (y) with respect to the independent variable (x)

What are the units of f’(x)?

units of y/units of x

What sentence can be used to describe a derivative in context?

The dependent variable (y) is inc/dec at a rate of f’(x) (with units) at independent variable (x).

• Inc/Dec depends on whether f’(x) is positive or negative

Position

• Describes location

• Positive: to the right of the origin

• Negative: to the left of the origin

Velocity

• Derivative of position: p’(t)

• Tells you the direction that the particle is moving

• Positive: Up/Right

• Negative: Down/Left

• 0: At rest

Acceleration

• Derivative of velocity v’(t) & second derivative of position p’’(t)

Speed

Iv(t)I

• Compare acceleration and velocity to see whether a particle is speeding or slowing

• Same sign: Speeding

• Different sign: Slowing

Related Rates

1. Draw a picture

2. Write a formula

3. Take derivative of each side w/ respect to time

4. Plug in everything you know and solve for the remaining variable

Tangent Line Formula

y= f’(a)(x-a)+f(a)

L’Hospital’s Rule