AP Calculus AB and BC · Exam guide

AP Calculus Related Rates and Optimization: A Setup Routine That Earns the Points

Related rates and optimization are the two applied derivative questions. Both are mostly setup: draw the situation, write an equation, and differentiate or find a maximum. This guide gives a routine for each and a checked example.

  • Draw and label the situation. Give each changing quantity a variable.
  • Write the equation that relates the variables. Do not substitute numbers yet.
  • Differentiate with respect to time, using the chain rule.
  • Substitute the values at the moment in question, including any values you need to find from the equation.
  • Solve for the unknown rate.
  • Answer in a sentence with units, and say whether the quantity is increasing or decreasing.

A 13-foot ladder leans against a wall. The bottom slides away from the wall at 2 feet per second. How fast is the top sliding down the wall when the bottom is 5 feet from the wall?

Let \(x\) be the distance from the wall to the bottom, and \(y\) the height of the top. Then \(x^2 + y^2 = 169\). Differentiating, \(2x\dfrac{dx}{dt} + 2y\dfrac{dy}{dt} = 0\), so \(\dfrac{dy}{dt} = -\dfrac{x}{y}\dfrac{dx}{dt}\). When \(x = 5\), \(y = 12\). So \(\dfrac{dy}{dt} = -\dfrac{5}{12}(2) = -\dfrac{5}{6}\). The top is sliding down the wall at \(\dfrac{5}{6}\) foot per second.

Does not earn the pointsdy/dt = 5/6.

It has the right size but no sign and no units, and it never shows the relationship.

Earns the pointsx^2 + y^2 = 169, so 2x dx/dt + 2y dy/dt = 0. With x = 5 and y = 12, dy/dt = -(5/12)(2) = -5/6 feet per second, so the top is sliding down at 5/6 feet per second.

It shows the equation, the derivative, the values, and the answer in a sentence with units.

Optimization: a routine

  • Write the quantity to maximize or minimize, and the constraint.
  • Use the constraint to write the quantity in one variable.
  • State the domain from the problem.
  • Find critical points, then justify the answer with a sign change of the derivative or with the candidates test on a closed interval.

A farmer has 24 meters of fence for a rectangular pen against a straight wall, so only three sides need fence. What dimensions give the greatest area?

Let \(x\) be the length of each of the two sides perpendicular to the wall. The side parallel to the wall is \(24 - 2x\). Area \(A(x) = x(24 - 2x) = 24x - 2x^2\) for \(0 \le x \le 12\). \(A'(x) = 24 - 4x = 0\) at \(x = 6\). The candidates are \(A(0) = 0\), \(A(6) = 72\), and \(A(12) = 0\). The greatest area is 72 square meters, with sides of 6 meters and 12 meters.

Common mistakes

  • Substituting numbers before differentiating, which turns the changing quantity into a constant.
  • Mixing up which variable is changing and which is a constant, such as the ladder length.
  • Not justifying that a critical point gives a maximum or minimum.
  • Leaving out units or the direction (increasing or decreasing).

Frequently asked questions

Should I substitute before or after differentiating?

After. Substituting early freezes the variables that change.

How do I justify an absolute maximum?

State the domain and compare the values at critical points and endpoints, or show that the derivative changes sign from positive to negative and there is only one critical point.

Does a negative rate matter?

Yes. A negative sign means the quantity is decreasing, and you should say so in words.

Are these in both AB and BC?

Yes.

Where can I find the official scoring rules?

Exam formats change a little from year to year, so check the current course and exam description on the College Board site before test day.

Keep practicing

Every lesson in the AP Calculus courses ends with practice problems and an Exam focus box that shows how the topic is tested, and every unit has a 10-problem practice set with worked answers. These free lessons also build the skills the free-response section tests:

See the AP Calculus AB and BC course Free lessons and lesson list See the AP Calculus BC course

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