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 9 · Topic 9.5 · BC Only

Integrating Vector-Valued Functions

Integrate components and use initial vectors to determine a particular position function.

1. Topic Focus

Represent planar motion parametrically and with vectors, then analyze polar derivatives and areas.

This topic: Integrate components and use initial vectors to determine a particular position function.

2. Key Relationship

\(\int\langle f,g\rangle dt=\left\langle\int fdt,\int gdt\right\rangle\)

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

3. Visual Connection

Vector-valued motionDifferentiate or integrate components while keeping position, velocity, acceleration, and their constants distinct.

4. Worked Example

Integrate acceleration to velocity and velocity to position, applying the matching initial condition at each stage.

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

5. Concept Development

BC-only topic. Integration of a vector-valued function is performed component by component. If \(\mathbf F'(t)=\mathbf f(t)\), then \(\mathbf F\) is a vector antiderivative of \(\mathbf f\).

Indefinite integrals

\(\int\langle f(t),g(t)\rangle\,dt=\left\langle\int f(t)\,dt,\int g(t)\,dt\right\rangle+\mathbf C,\)

where \(\mathbf C=\langle C_1,C_2\rangle\). Each component has its own integration constant. Writing only one scalar constant would incorrectly force the two components to shift by the same amount.

Definite integrals and the Fundamental Theorem

\(\int_a^b\langle f(t),g(t)\rangle\,dt=\left\langle\int_a^b f(t)\,dt,\int_a^b g(t)\,dt\right\rangle.\)

If \(\mathbf r'(t)=\mathbf v(t)\), then

\(\int_a^b\mathbf v(t)\,dt=\mathbf r(b)-\mathbf r(a).\)

The result is a displacement vector, not total distance. Distance requires the scalar integral \(\int_a^b\|\mathbf v(t)\|dt\), developed further in Topic 9.6.

Use initial conditions directly

The cleanest way to build the particular position function from velocity is

\(\boxed{\mathbf r(t)=\mathbf r(t_0)+\int_{t_0}^{t}\mathbf v(s)\,ds}.\)

The dummy variable \(s\) prevents confusion between the upper limit \(t\) and the integration variable. This form automatically satisfies \(\mathbf r(t_0)\) and avoids solving separately for constants.

From acceleration to position

Acceleration must be integrated twice:

\(\mathbf v(t)=\mathbf v(t_0)+\int_{t_0}^{t}\mathbf a(s)\,ds,\qquad \mathbf r(t)=\mathbf r(t_0)+\int_{t_0}^{t}\mathbf v(s)\,ds.\)

An initial velocity determines the constant introduced by the first integration; an initial position determines the constant introduced by the second. Position data alone cannot determine velocity after integrating acceleration once.

Net change from a table

When velocity components are tabulated, approximate each component integral separately. For trapezoidal integration,

\(\Delta\mathbf r\approx\left\langle T[v_x],T[v_y]\right\rangle.\)

Use the actual time widths, preserve component signs, and add the resulting displacement to the initial position if a final position is requested.

Verification and units

Differentiate the proposed position to recover velocity.
If acceleration was given, differentiate velocity to recover acceleration.
Substitute every initial condition.
Check units: integrating acceleration over time gives velocity; integrating velocity gives position or displacement.

Linearity

For vector functions \(\mathbf f,\mathbf g\) and scalar \(c\), integration preserves sums and constant multiples:

\(\int(\mathbf f+\mathbf g)\,dt=\int\mathbf f\,dt+\int\mathbf g\,dt,\qquad \int c\mathbf f\,dt=c\int\mathbf f\,dt.\)

These properties follow componentwise from ordinary scalar integration.

6. Detailed Worked Example and Error Check

Example 1: An indefinite vector integral

Integrate \(\mathbf f(t)=\langle3t^2,2\cos t\rangle\):

\(\int\mathbf f(t)\,dt=\langle t^3,2\sin t\rangle+\langle C_1,C_2\rangle.\)

Differentiating returns \(\langle3t^2,2\cos t\rangle\).

Example 2: Recover position from velocity

Suppose \(\mathbf v(t)=\langle2t,3t^2\rangle\) and \(\mathbf r(1)=\langle4,-2\rangle\). Then

\(\begin{aligned}\mathbf r(t)&=\langle4,-2\rangle+\int_1^t\langle2s,3s^2\rangle ds\\&=\langle4,-2\rangle+\langle t^2-1,t^3-1\rangle\\&=\langle t^2+3,t^3-3\rangle.\end{aligned}\)

Example 3: Integrate acceleration twice

Let \(\mathbf a(t)=\langle6t,-2\rangle\), \(\mathbf v(0)=\langle1,4\rangle\), and \(\mathbf r(0)=\langle2,-1\rangle\). First,

\(\mathbf v(t)=\langle1,4\rangle+\int_0^t\langle6s,-2\rangle ds=\langle3t^2+1,4-2t\rangle.\)

Then

\(\mathbf r(t)=\langle2,-1\rangle+\int_0^t\langle3s^2+1,4-2s\rangle ds=\langle t^3+t+2,-t^2+4t-1\rangle.\)

Example 4: Initial time is not zero

Suppose \(\mathbf a(t)=\langle2,e^t\rangle\), \(\mathbf v(1)=\langle0,e\rangle\), and \(\mathbf r(1)=\langle3,0\rangle\). Then

\(\mathbf v(t)=\langle2t-2,e^t\rangle,\)

\(\mathbf r(t)=\left\langle3+(t-1)^2,e^t-e\right\rangle.\)

Substituting \(t=1\) verifies both initial vectors.

Example 5: A definite integral gives displacement

For \(\mathbf v(t)=\langle t-1,t^2-1\rangle\) on \([0,2]\),

\(\Delta\mathbf r=\int_0^2\mathbf v(t)dt=\left\langle0,\frac23\right\rangle.\)

The horizontal displacement is zero even though the particle may have moved horizontally during the interval.

Example 6: Approximate displacement from a table

At \(t=0,1,3\), suppose velocity is \(\langle2,0\rangle,\langle4,-2\rangle,\langle0,2\rangle\). Applying the trapezoidal rule to each component gives

\(\Delta\mathbf r\approx\left\langle1\left(\frac{2+4}{2}\right)+2\left(\frac{4+0}{2}\right),\;1\left(\frac{0-2}{2}\right)+2\left(\frac{-2+2}{2}\right)\right\rangle=\langle7,-1\rangle.\)

Example 7: Determine the vector constant

Given \(\mathbf r'(t)=\langle\cos t,2t\rangle\) and \(\mathbf r(0)=\langle1,-2\rangle\), a general antiderivative is

\(\mathbf r(t)=\langle\sin t,t^2\rangle+\langle C_1,C_2\rangle.\)

The initial condition gives \(\langle C_1,C_2\rangle=\langle1,-2\rangle\), so \(\mathbf r(t)=\langle\sin t+1,t^2-2\rangle\).

Example 8: Same velocity, translated paths

If \(\mathbf v(t)=\langle1,2t\rangle\), then every possible position function has the form

\(\mathbf r(t)=\langle t,t^2\rangle+\mathbf C.\)

Different initial positions select different constant vectors and translate the entire path without changing its velocity at any time.

Common errors

Using one scalar constant instead of a constant vector.
Forgetting the initial velocity when integrating acceleration.
Using initial position to determine the velocity constant.
Integrating acceleration only once when position is requested.
Confusing \(\int\mathbf v\,dt\) with \(\int\|\mathbf v\|dt\).
Dropping negative signs in a component integral.
Applying a numerical rule to vector magnitudes when displacement components are requested.
Failing to verify all initial conditions.

7. AP Reasoning Routine

Keep the parameter visible until the requested quantity is formed, track orientation and speed, and choose polar bounds from the traced region.

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

Integrate each vector-valued function and apply the stated initial conditions.
(a) Find \(\int\langle4t^3,e^t\rangle dt\).
(b) Given \(\mathbf v(t)=\langle2t,\cos t\rangle\) and \(\mathbf r(0)=\langle1,3\rangle\), find \(\mathbf r(t)\).
(c) Given \(\mathbf v(t)=\langle3t^2,-2\rangle\) and \(\mathbf r(1)=\langle0,4\rangle\), find \(\mathbf r(t)\).
(d) Given \(\mathbf a(t)=\langle2,6t\rangle\) and \(\mathbf v(0)=\langle-1,2\rangle\), find \(\mathbf v(t)\).
(e) Add \(\mathbf r(0)=\langle4,-3\rangle\) to part (d) and find \(\mathbf r(t)\).
(f) Given \(\mathbf a(t)=\langle2,e^t\rangle\), \(\mathbf v(1)=\langle0,e\rangle\), and \(\mathbf r(1)=\langle3,0\rangle\), find velocity and position.
(g) Find the displacement generated by \(\mathbf v(t)=\langle t^2,2t-1\rangle\) on \(0\le t\le2\).
(h) At \(t=0,1,3\), velocity is \(\langle2,0\rangle,\langle4,-2\rangle,\langle0,2\rangle\). Use the trapezoidal rule to estimate displacement. If \(\mathbf r(0)=\langle1,2\rangle\), estimate \(\mathbf r(3)\).
(i) Verify that \(\mathbf r(t)=\langle\sin t+1,t^2-2\rangle\) solves \(\mathbf r'(t)=\langle\cos t,2t\rangle\), \(\mathbf r(0)=\langle1,-2\rangle\).
(j) Explain the difference between \(\int_a^b\mathbf v(t)dt\) and \(\int_a^b\|\mathbf v(t)\|dt\).

Check the solution

(a) \(\langle t^4,e^t\rangle+\langle C_1,C_2\rangle\).
(b) \(\mathbf r(t)=\langle1,3\rangle+\int_0^t\langle2s,\cos s\rangle ds=\langle t^2+1,\sin t+3\rangle\).
(c) \(\mathbf r(t)=\langle0,4\rangle+\int_1^t\langle3s^2,-2\rangle ds=\langle t^3-1,6-2t\rangle\).
(d) \(\mathbf v(t)=\langle-1,2\rangle+\int_0^t\langle2,6s\rangle ds=\langle2t-1,3t^2+2\rangle\).
(e) \(\mathbf r(t)=\langle4,-3\rangle+\int_0^t\langle2s-1,3s^2+2\rangle ds=\langle t^2-t+4,t^3+2t-3\rangle\).
(f) \(\mathbf v(t)=\langle2t-2,e^t\rangle\) and \(\mathbf r(t)=\langle3+(t-1)^2,e^t-e\rangle\).
(g) \(\Delta\mathbf r=\int_0^2\langle t^2,2t-1\rangle dt=\langle8/3,2\rangle\).
(h) Componentwise trapezoidal integration gives \(\Delta\mathbf r\approx\langle7,-1\rangle\), so \(\mathbf r(3)\approx\langle8,1\rangle\).
(i) Differentiation gives \(\langle\cos t,2t\rangle\), and substitution at \(t=0\) gives \(\langle1,-2\rangle\), so both requirements hold.
(j) The first integral is the displacement vector and can contain cancellation in each component. The second integrates nonnegative speed and gives scalar total distance traveled.