AP Calculus AB Unit 4: Contextual Applications
Applications of Differentiation
Motion, Related Rates, and Approximations
1. Straight-Line Motion
Motion along a line (particle motion) is a primary application of derivatives.
Position: x(t) or s(t) Speed: |v(t)| (Magnitude of velocity)
Velocity: v(t) = s′ (t) Displacement: ∆s = s(b) − s(a)
Acceleration: a(t) = v ′ (t) = s′′ (t)
Rb
Total Distance: a |v(t)| dt
Key Concept
Speeding Up vs. Slowing Down:
• Speeding Up: v(t) and a(t) have the SAME sign (++, - -).
• Slowing Down: v(t) and a(t) have DIFFERENT signs (+-, -+).
2. Related Rates
Finding the rate of change of one quantity by relating it to other quantities whose rates of change are
known. All variables are implicit functions of time t.
Strategy for Related Rates
1. Draw a picture and label variables (Equation, Given, Find).
2. Write the equation relating the variables (Pythagorean theorem, Volume, Trig).
3. Differentiate implicitly with respect to time t ( dt
d
).
4. Substitute known values after differentiating.
5. Solve for the unknown rate.
Common Pitfall
Do not plug in numbers for variables that are changing until AFTER you take the derivative!
Only plug in constants beforehand.
3. Linearization (Tangent Line Approx.)
Using the tangent line at a point (a, f (a)) to approximate values of the function near a.
1
Applications of Differentiation
Motion, Related Rates, and Approximations
1. Straight-Line Motion
Motion along a line (particle motion) is a primary application of derivatives.
Position: x(t) or s(t) Speed: |v(t)| (Magnitude of velocity)
Velocity: v(t) = s′ (t) Displacement: ∆s = s(b) − s(a)
Acceleration: a(t) = v ′ (t) = s′′ (t)
Rb
Total Distance: a |v(t)| dt
Key Concept
Speeding Up vs. Slowing Down:
• Speeding Up: v(t) and a(t) have the SAME sign (++, - -).
• Slowing Down: v(t) and a(t) have DIFFERENT signs (+-, -+).
2. Related Rates
Finding the rate of change of one quantity by relating it to other quantities whose rates of change are
known. All variables are implicit functions of time t.
Strategy for Related Rates
1. Draw a picture and label variables (Equation, Given, Find).
2. Write the equation relating the variables (Pythagorean theorem, Volume, Trig).
3. Differentiate implicitly with respect to time t ( dt
d
).
4. Substitute known values after differentiating.
5. Solve for the unknown rate.
Common Pitfall
Do not plug in numbers for variables that are changing until AFTER you take the derivative!
Only plug in constants beforehand.
3. Linearization (Tangent Line Approx.)
Using the tangent line at a point (a, f (a)) to approximate values of the function near a.
1
