AP Course

AP Calculus AB/BC

Study the complete College Board sequence for AP Calculus AB and BC, from limits through infinite series.

Choose an official unit and topic to open its lecture, concept check, or focused practice.

Lessons

1.1 Introducing Calculus: Can Change Occur at an Instant?1.2 Defining Limits and Using Limit Notation1.3 Estimating Limit Values from Graphs1.4 Estimating Limit Values from Tables1.5 Determining Limits Using Algebraic Properties of Limits1.6 Determining Limits Using Algebraic Manipulation1.7 Selecting Procedures for Determining Limits1.8 Determining Limits Using the Squeeze Theorem1.9 Connecting Multiple Representations of Limits1.10 Exploring Types of Discontinuities1.11 Defining Continuity at a Point1.12 Confirming Continuity over an Interval1.13 Removing Discontinuities1.14 Connecting Infinite Limits and Vertical Asymptotes1.15 Connecting Limits at Infinity and Horizontal Asymptotes1.16 Working with the Intermediate Value Theorem (IVT)2.1 Defining Average and Instantaneous Rates of Change at a Point2.2 Defining the Derivative of a Function and Using Derivative Notation2.3 Estimating Derivatives of a Function at a Point2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist2.5 Applying the Power Rule2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple2.7 Derivatives of cos x, sin x, e^x, and ln x2.8 The Product Rule2.9 The Quotient Rule2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions3.1 The Chain Rule3.2 Implicit Differentiation3.3 Differentiating Inverse Functions3.4 Differentiating Inverse Trigonometric Functions3.5 Selecting Procedures for Calculating Derivatives3.6 Calculating Higher-Order Derivatives4.1 Interpreting the Meaning of the Derivative in Context4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration4.3 Rates of Change in Applied Contexts Other Than Motion4.4 Introduction to Related Rates4.5 Solving Related Rates Problems4.6 Approximating Values of a Function Using Local Linearity and Linearization4.7 Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms5.1 Using the Mean Value Theorem5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points5.3 Determining Intervals on Which a Function Is Increasing or Decreasing5.4 Using the First Derivative Test to Determine Relative (Local) Extrema5.5 Using the Candidates Test to Determine Absolute (Global) Extrema5.6 Determining Concavity of Functions over Their Domains5.7 Using the Second Derivative Test to Determine Extrema5.8 Sketching Graphs of Functions and Their Derivatives5.9 Connecting a Function, Its First Derivative, and Its Second Derivative5.10 Introduction to Optimization Problems5.11 Solving Optimization Problems5.12 Exploring Behaviors of Implicit Relations6.1 Exploring Accumulations of Change6.2 Approximating Areas with Riemann Sums6.3 Riemann Sums, Summation Notation, and Definite Integral Notation6.4 The Fundamental Theorem of Calculus and Accumulation Functions6.5 Interpreting the Behavior of Accumulation Functions Involving Area6.6 Applying Properties of Definite Integrals6.7 The Fundamental Theorem of Calculus and Definite Integrals6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation6.9 Integrating Using Substitution6.10 Integrating Functions Using Long Division and Completing the Square6.11 Integrating Using Integration by Parts6.12 Using Linear Partial Fractions6.13 Evaluating Improper Integrals6.14 Selecting Techniques for Antidifferentiation7.1 Modeling Situations with Differential Equations7.2 Verifying Solutions for Differential Equations7.3 Sketching Slope Fields7.4 Reasoning Using Slope Fields7.5 Approximating Solutions Using Euler’s Method7.6 Finding General Solutions Using Separation of Variables7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables7.8 Exponential Models with Differential Equations7.9 Logistic Models with Differential Equations8.1 Finding the Average Value of a Function on an Interval8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts8.4 Finding the Area Between Curves Expressed as Functions of x8.5 Finding the Area Between Curves Expressed as Functions of y8.6 Finding the Area Between Curves That Intersect at More Than Two Points8.7 Volumes with Cross Sections: Squares and Rectangles8.8 Volumes with Cross Sections: Triangles and Semicircles8.9 Volume with Disc Method: Revolving Around the x- or y-Axis8.10 Volume with Disc Method: Revolving Around Other Axes8.11 Volume with Washer Method: Revolving Around the x- or y-Axis8.12 Volume with Washer Method: Revolving Around Other Axes8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled9.1 Defining and Differentiating Parametric Equations9.2 Second Derivatives of Parametric Equations9.3 Finding Arc Lengths of Curves Given by Parametric Equations9.4 Defining and Differentiating Vector-Valued Functions9.5 Integrating Vector-Valued Functions9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions9.7 Defining Polar Coordinates and Differentiating in Polar Form9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve9.9 Finding the Area of the Region Bounded by Two Polar Curves10.1 Defining Convergent and Divergent Infinite Series10.2 Working with Geometric Series10.3 The nth Term Test for Divergence10.4 Integral Test for Convergence10.5 Harmonic Series and p-Series10.6 Comparison Tests for Convergence10.7 Alternating Series Test for Convergence10.8 Ratio Test for Convergence10.9 Determining Absolute or Conditional Convergence10.10 Alternating Series Error Bound10.11 Finding Taylor Polynomial Approximations of Functions10.12 Lagrange Error Bound10.13 Radius and Interval of Convergence of Power Series10.14 Finding Taylor or Maclaurin Series for a Function10.15 Representing Functions as Power Series

Quizzes

Practice Problems AP formula notes, graph references, and practice sets will be added here.

AP Calculus AB/BC · Unit 4 · Topic 4.5

Solving Related Rates Problems

Model, differentiate, substitute instantaneous data, solve, and interpret a requested rate.

1. Topic Focus

Interpret derivatives as rates in context, connect motion quantities, solve related-rate models, linearize, and evaluate indeterminate limits.

This topic: Model, differentiate, substitute instantaneous data, solve, and interpret a requested rate.

2. Key Relationship

\(\text{model}\to\frac d{dt}\to\text{substitute}\to\text{solve}\)

Read every symbol with its domain, direction, units, and hypotheses before applying the relationship.

3. Visual Connection

Choose the relationship before differentiatingRight triangles, similar figures, volume formulas, and trigonometric ratios organize the major related-rates problem families.

4. Worked Example

For an expanding circle, dA/dt=2πr·dr/dt.

Write the governing relationship first, carry out the algebra cleanly, and finish with a sentence that answers the mathematical question.

5. Concept Development

1. Decode the Prompt before Calculating

Underline every changing quantity, instantaneous value, given rate, and requested rate. Translate verbal directions into signed derivatives:

Phrase	Typical derivative translation
increasing at \(k\)	\(dq/dt=k\)
decreasing at \(k\)	\(dq/dt=-k\)
moving away from a reference point	positive distance rate
moving toward a reference point	negative distance rate
fixed length, height, or volume	the corresponding derivative is zero

Record units immediately; they help distinguish a value from a rate.

2. The Full Solution Pipeline

Draw and label a diagram when geometry is involved.
Define changing variables as functions of time.
List known values, signed rates, and the requested rate.
Write one equation relating the changing quantities.
Differentiate the equation with respect to time.
Use the original equation to find missing instantaneous values.
Substitute values and rates into the differentiated equation.
Solve and state the answer with sign, units, and context.

3. A Diagram Is a Snapshot, Not a Frozen Model

A diagram usually represents the system at the requested instant. Labels such as \(x(t)\), \(y(t)\), and \(r(t)\) remain variable even if the prompt later specifies \(x=5\) or \(r=3\). Fixed dimensions may be labeled with constants. Mark right angles, corresponding sides, and the direction of motion.

4. Choose the Smallest Useful Set of Variables

Define only quantities needed to connect the given and requested rates. Extra variables create extra derivatives. For a plane at constant altitude, use horizontal distance and line-of-sight distance; the fixed altitude can remain a constant. For a conical tank, similar triangles may eliminate radius before differentiation.

5. Match the Situation to a Relationship

Problem family	Useful relationship
ladder, line of sight, perpendicular travel	\(x^2+y^2=z^2\)
circle or sphere	\(A=\pi r^2\), \(V=\frac43\pi r^3\)
cylinder or cone	\(V=\pi r^2h\), \(V=\frac13\pi r^2h\)
shadows or conical containers	similar-triangle proportions
angle of elevation	\(\tan\theta=\text{opposite}/\text{adjacent}\)
changing rectangle	\(A=\ell w\)

6. Apply the Correct Differentiation Rule

Related-rates equations frequently require more than the power rule:

\(\frac{d}{dt}(x^2+y^2)=2x\frac{dx}{dt}+2y\frac{dy}{dt},\)

\(\frac{d}{dt}(\ell w)=\ell\frac{dw}{dt}+w\frac{d\ell}{dt},\)

\(\frac{d}{dt}(\tan\theta)=\sec^2\theta\frac{d\theta}{dt}.\)

Every changing factor must contribute its derivative.

7. Substitute Values Only after Differentiation

Instantaneous values describe one moment, whereas the model must remain valid through nearby moments for a rate to exist. Substituting \(x=5\) before differentiating incorrectly makes \(x\) constant. Keep variables symbolic until the rate relationship has been formed.

8. Recover Missing Instantaneous Values

Use the original relationship, not the differentiated one, to find an unstated length or angle value. A 13-foot ladder with base 5 feet from the wall has height

\(y=\sqrt{13^2-5^2}=12\text{ ft}.\)

Choose the physically meaningful root: dimensions and distances are nonnegative.

9. Right-Triangle Distance Problems

When \(x^2+y^2=z^2\), decide which side is fixed and which sides change. If all three can change, differentiation gives

\(x\frac{dx}{dt}+y\frac{dy}{dt}=z\frac{dz}{dt}.\)

A fixed hypotenuse makes \(dz/dt=0\). For two objects moving perpendicularly, both leg rates may be nonzero.

10. Similar Triangles Remove Extra Variables

In a conical tank or shadow problem, identify corresponding sides and write a proportion. If a cone maintains \(r/h=k\), substitute \(r=kh\) into its volume formula before differentiating. This leaves one changing dimension instead of two unrelated-looking rates.

11. Shadow Problems Contain Two Different Lengths

Let \(x\) be the person's distance from the light, \(y\) the shadow length, and \(z=x+y\) the distance from the light to the shadow tip. The prompt may ask for \(dy/dt\) or \(dz/dt\); these are different rates. Translate the requested motion before solving.

12. Angle Problems Use Radians

For an angle of elevation, a tangent relationship often avoids the hypotenuse:

\(\tan\theta=\frac{h}{x}.\)

Differentiate with respect to time and include \(d\theta/dt\). Angular rates from calculus are measured in radians per unit time. If one distance is fixed, its derivative is zero.

13. Multiple Input Rates Can Compete

When several dimensions change, retain every product-rule term. For \(A=\ell w\), an increasing length and decreasing width can make area increase, decrease, or remain momentarily constant. The net sign is determined only after both contributions are combined.

14. Verify the Final Result

Sign: Does it agree with the chosen direction and physical motion?
Units: Does a length, area, volume, or angle rate have the correct dimensions?
Magnitude: Is the result plausible relative to the given rates and geometry?
Instant: Did you use values from the requested moment?
Question: Did you solve for the requested derivative rather than a nearby quantity?

15. Common AP Solution Errors

Using a diagram but never defining its variables.
Dropping chain-rule or product-rule factors.
Confusing a decreasing rate with a positive speed.
Failing to use similar triangles before differentiating a cone or shadow model.
Substituting instantaneous values too early.
Using degrees instead of radians for \(d\theta/dt\).
Finding the shadow-length rate when the problem asks for the tip rate.
Providing correct algebra without a contextual conclusion and units.

6. Detailed Worked Example and Error Check

Example 1: Sliding ladder. A 13-foot ladder rests against a wall. Let \(x\) be the base's distance from the wall and \(y\) the top's height. Then

\(x^2+y^2=169\quad\Longrightarrow\quad x\frac{dx}{dt}+y\frac{dy}{dt}=0.\)

When \(x=5\), the original equation gives \(y=12\). If \(dx/dt=2\) feet per second,

\(5(2)+12\frac{dy}{dt}=0\quad\Longrightarrow\quad\boxed{\frac{dy}{dt}=-\frac56\text{ ft/s}}.\)

The ladder top slides downward at \(5/6\) foot per second.

Example 2: Inflating sphere. A spherical balloon receives air at \(dV/dt=72\pi\) cubic centimeters per second. Since

\(V=\frac43\pi r^3\quad\Longrightarrow\quad\frac{dV}{dt}=4\pi r^2\frac{dr}{dt},\)

when \(r=3\) centimeters,

\(72\pi=4\pi(3)^2\frac{dr}{dt}\quad\Longrightarrow\quad\boxed{\frac{dr}{dt}=2\text{ cm/s}}.\)

Example 3: Water in a similar conical tank. A cone maintains \(r/h=1/3\), so \(r=h/3\). Eliminate radius from the volume equation:

\(V=\frac13\pi r^2h=\frac{\pi}{27}h^3.\)

If water enters at \(dV/dt=4\pi\) cubic centimeters per second, then

\(\frac{dV}{dt}=\frac{\pi h^2}{9}\frac{dh}{dt}.\)

At \(h=3\) centimeters, \(4\pi=\pi\,dh/dt\), so \(\boxed{dh/dt=4\text{ cm/s}}\).

Example 4: Person and shadow tip. A 6-foot person walks away from a 12-foot lamp at 4 feet per second. Let \(x\) be the person's distance from the lamp and \(y\) the shadow length. Similar triangles give

\(\frac{12}{x+y}=\frac6y\quad\Longrightarrow\quad y=x.\)

Thus \(dy/dt=dx/dt=4\) feet per second. The shadow tip is \(z=x+y\), so

\(\boxed{\frac{dz}{dt}=\frac{dx}{dt}+\frac{dy}{dt}=8\text{ ft/s}}.\)

The shadow length grows at 4 feet per second, while the tip moves away from the lamp at 8 feet per second.

Example 5: Camera angle of elevation. A camera is 500 feet from a launch point. A rocket rises vertically at \(dh/dt=200\) feet per second. Let \(\theta\) be the camera's angle of elevation:

\(\tan\theta=\frac{h}{500}\quad\Longrightarrow\quad\sec^2\theta\frac{d\theta}{dt}=\frac1{500}\frac{dh}{dt}.\)

When \(h=500\), \(\tan\theta=1\), so \(\sec^2\theta=2\). Therefore

\(2\frac{d\theta}{dt}=\frac{200}{500}\quad\Longrightarrow\quad\boxed{\frac{d\theta}{dt}=0.2\text{ rad/s}}.\)

7. AP Reasoning Routine

Name variables and units, write the relationship before differentiating, substitute values at the correct time, and interpret the sign in context.

Identify the representation and requested quantity.
State the rule or theorem and verify its conditions.
Keep exact values until the final requested approximation.
Interpret sign, units, interval, and context.

AP Checkpoint

Solve each related-rates problem with defined variables, a differentiated relationship, units, and a contextual conclusion.
(a) A 10-foot ladder has base distance \(x\) and height \(y\). The base moves away from the wall at 2 feet per second. Find \(dy/dt\) when \(x=6\) feet.
(b) A sphere's volume increases at \(100\pi\) cubic centimeters per second. Find \(dr/dt\) when \(r=5\) centimeters.
(c) A conical tank maintains \(r=h/2\). Water enters at \(12\pi\) cubic centimeters per second. Find \(dh/dt\) when \(h=4\) centimeters.
(d) A 6-foot person walks away from a 15-foot lamp at 3 feet per second. Find both the shadow-length rate and the rate at which the shadow tip moves away from the lamp.
(e) A camera is 300 feet from a launch point. A balloon rises at 60 feet per second. Find the angle-of-elevation rate when the balloon is 300 feet high.
(f) Two cyclists leave the same point on perpendicular roads. At one instant they are 6 and 8 miles from the start and moving away at 3 and 4 miles per hour. How fast is the distance between them changing?
(g) Explain why instantaneous length values must be substituted after differentiating even when the requested moment is clearly specified.
(h) Give one sign check and one unit check for the answer in part (a).

Check the solution

In part (a), \(x^2+y^2=100\), so \(y=8\) when \(x=6\). From \(x\,dx/dt+y\,dy/dt=0\), \(dy/dt=-6(2)/8=-3/2\) feet per second; the top moves downward. In part (b), \(dV/dt=4\pi r^2\,dr/dt\), so \(100\pi=4\pi(25)\,dr/dt\) and \(dr/dt=1\) centimeter per second. In part (c), \(V=\frac13\pi(h/2)^2h=\pi h^3/12\), so \(dV/dt=(\pi h^2/4)\,dh/dt\). At \(h=4\), \(12\pi=4\pi\,dh/dt\), giving \(dh/dt=3\) centimeters per second. In part (d), similar triangles give \(15/(x+y)=6/y\), so \(9y=6x\) and \(dy/dt=(2/3)(3)=2\) feet per second. Since \(z=x+y\), the tip rate is \(dz/dt=3+2=5\) feet per second. In part (e), \(\tan\theta=h/300\), so \(\sec^2\theta\,d\theta/dt=(1/300)dh/dt\). At \(h=300\), \(\sec^2\theta=2\), giving \(d\theta/dt=0.1\) radian per second. In part (f), \(x^2+y^2=z^2\), so \(z=10\) and \(z\,dz/dt=x\,dx/dt+y\,dy/dt=6(3)+8(4)=50\); hence \(dz/dt=5\) miles per hour. In part (g), those values hold only at one instant; early substitution falsely turns changing quantities into constants and removes their derivatives. In part (h), the top height should decrease, so \(dy/dt\) should be negative, and a height rate must have feet-per-second units.