United States · College BoardSyllabus
Calculus syllabus, dot point by dot point
Every dot point in the United States Calculussyllabus, with a focused answer for each one. Click any dot point for a worked explainer, past exam questions, and links to related dot points. Written by Claude Opus 4.8, Anthropic's latest AI.
Unit 1: Limits and Continuity
Module overview →- How can a quantity have a rate of change at a single instant, when change seems to require an interval of time?Topic 1.1 Introducing Calculus - Can Change Occur at an Instant?: understand how the idea of an instantaneous rate of change motivates the limit, and how average rates of change over shrinking intervals approach it.9 min answer →
- How do the graphical, numerical, and algebraic views of a limit fit together into one consistent answer?Topic 1.9 Connecting Multiple Representations of Limits: translate among graphical, numerical, analytical and verbal representations of a limit and confirm they agree.8 min answer →
- What exactly must be true for a function to be continuous at a single point?Topic 1.11 Defining Continuity at a Point: state and apply the three-part definition of continuity at a point and test functions against it.8 min answer →
- What does it mean for a function to be continuous over an entire interval, and which familiar functions qualify?Topic 1.12 Confirming Continuity over an Interval: determine intervals on which a function is continuous, using one-sided continuity at endpoints and the continuity of standard function families.8 min answer →
- What does it mean for a function to approach a value, and how do we write that idea precisely?Topic 1.2 Defining Limits and Using Limit Notation: express the limit of a function using correct notation, including one-sided limits, and interpret what a limit says about the behavior of a function near a point.9 min answer →
- How do you read the limit of a function at a point directly from its graph, even where the function is undefined?Topic 1.3 Estimating Limit Values from Graphs: use a graph to estimate one-sided and two-sided limits, including cases where the function value differs from the limit or does not exist.8 min answer →
- How can a table of function values let you estimate a limit you cannot evaluate directly?Topic 1.4 Estimating Limit Values from Tables: use a table of values approaching a point from both sides to estimate one-sided and two-sided limits.8 min answer →
- What does an infinite limit tell you about a function, and how does it locate a vertical asymptote?Topic 1.14 Connecting Infinite Limits and Vertical Asymptotes: use infinite limits to identify vertical asymptotes and describe behavior near them with correct sign analysis.8 min answer →
- How does continuity guarantee that a function must hit every value between its endpoints?Topic 1.16 Working with the Intermediate Value Theorem (IVT): state the hypotheses of the IVT and use it to guarantee the existence of a value or a root on a closed interval.8 min answer →
- What happens to a function as the input grows without bound, and how does that reveal a horizontal asymptote?Topic 1.15 Connecting Limits at Infinity and Horizontal Asymptotes: evaluate limits as x approaches plus or minus infinity and use them to identify horizontal asymptotes, especially for rational functions.9 min answer →
- When direct substitution gives the indeterminate form 0/0, how do you rewrite the function to find the limit?Topic 1.6 Determining Limits Using Algebraic Manipulation: resolve indeterminate forms by factoring and cancelling, rationalizing, combining fractions, or using known trigonometric limits.9 min answer →
- How do the limit laws let you break a complicated limit into simple pieces you can evaluate?Topic 1.5 Determining Limits Using Algebraic Properties of Limits: apply the limit laws (sum, difference, product, quotient, constant multiple, power) and direct substitution to evaluate limits.8 min answer →
- When a function has a hole, how do you redefine it at that point to make it continuous?Topic 1.13 Removing Discontinuities: recognize a removable discontinuity and define or redefine the function value to make it continuous.8 min answer →
- Given any limit, how do you decide which method - substitution, algebra, a table, or a graph - to use?Topic 1.7 Selecting Procedures for Determining Limits: choose an efficient strategy for a given limit, recognizing which technique fits the form of the function.8 min answer →
- How can you find a limit by trapping a function between two others that share the same limit?Topic 1.8 Determining Limits Using the Squeeze Theorem: apply the squeeze (sandwich) theorem to evaluate limits of functions bounded between two functions with a common limit.8 min answer →
- What are the different ways a function can fail to be continuous, and how do you tell them apart?Topic 1.10 Exploring Types of Discontinuities: classify discontinuities as removable (point/hole), jump, or infinite (asymptotic), using limits.8 min answer →
Unit 10: Infinite Sequences and Series
Module overview →- How do you bound the error when approximating an alternating series by a partial sum?Topic 10.10 Alternating Series Error Bound: bound the error of a partial-sum approximation of an alternating series by the magnitude of the first omitted term (BC).8 min answer →
- How does the alternating series test establish convergence of a series whose signs alternate?Topic 10.7 Alternating Series Test for Convergence: use the alternating series test (terms decreasing in magnitude to zero) to conclude convergence (BC).8 min answer →
- What does it mean for an infinite series to converge, in terms of its partial sums?Topic 10.1 Defining Convergent and Divergent Infinite Series: define the convergence of an infinite series as the limit of its sequence of partial sums (BC).9 min answer →
- What is the difference between absolute and conditional convergence?Topic 10.9 Determining Absolute or Conditional Convergence: classify a convergent series as absolutely or conditionally convergent by testing the series of absolute values (BC).8 min answer →
- How do you find the Taylor or Maclaurin series of a function, and what are the standard ones?Topic 10.14 Finding Taylor or Maclaurin Series for a Function: write the full Taylor or Maclaurin series of a function and recall the standard series for e^x, sin x, cos x and 1/(1-x) (BC).10 min answer →
- How do you build a Taylor polynomial that approximates a function near a point?Topic 10.11 Finding Taylor Polynomial Approximations of Functions: construct the Taylor (or Maclaurin) polynomial of a function about a center using its derivatives at that point (BC).10 min answer →
- When does a p-series converge, and why does the harmonic series diverge?Topic 10.5 Harmonic Series and p-Series: apply the p-series rule (converges iff p greater than 1) and recognize the harmonic series as the divergent p = 1 case (BC).8 min answer →
- How do you bound the error of a Taylor polynomial approximation using the Lagrange error bound?Topic 10.12 Lagrange Error Bound: bound the error of a Taylor polynomial approximation using the Lagrange form of the remainder (BC).9 min answer →
- How do you find the radius and interval of convergence of a power series?Topic 10.13 Radius and Interval of Convergence of Power Series: find the radius and interval of convergence of a power series using the ratio test and checking the endpoints (BC).10 min answer →
- How do you manipulate power series to represent new functions and evaluate hard integrals?Topic 10.15 Representing Functions as Power Series: manipulate known power series by substitution, multiplication, differentiation and integration to represent new functions and evaluate otherwise intractable integrals (BC).10 min answer →
- How do you decide convergence by comparing a series to a known benchmark series?Topic 10.6 Comparison Tests for Convergence: use the direct comparison test and the limit comparison test against a known p-series or geometric series (BC).9 min answer →
- How does an improper integral decide the convergence of a series with positive decreasing terms?Topic 10.4 Integral Test for Convergence: use the convergence of a related improper integral to decide convergence of a series with positive, decreasing terms (BC).9 min answer →
- How does the nth term test show a series diverges, and why can it never prove convergence?Topic 10.3 The nth Term Test for Divergence: use the limit of the terms to conclude divergence when the terms do not approach zero (BC).8 min answer →
- How does the ratio test use the limit of consecutive-term ratios to decide convergence?Topic 10.8 Ratio Test for Convergence: use the limit of the ratio of consecutive terms to decide absolute convergence or divergence, especially for factorials and exponentials (BC).9 min answer →
- When does a geometric series converge, and what is its sum?Topic 10.2 Working with Geometric Series: determine convergence of a geometric series by its common ratio and find its sum with the a over one-minus-r formula (BC).9 min answer →
Unit 2: Differentiation: Definition and Fundamental Properties
Module overview →- How does the average rate of change over an interval become the instantaneous rate of change at a point?Topic 2.1 Defining Average and Instantaneous Rates of Change at a Point: compute the average rate of change over an interval and define the instantaneous rate of change as the limit of average rates.8 min answer →
- How do the constant, sum, difference, and constant-multiple rules let you differentiate any polynomial term by term?Topic 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple: apply the basic linearity rules of differentiation to combine derivatives of individual terms.8 min answer →
- What is the formal definition of the derivative, and how do you write it in the different standard notations?Topic 2.2 Defining the Derivative of a Function and Using Derivative Notation: state the limit definition of the derivative, compute derivatives from the definition, and use standard derivative notation.9 min answer →
- How do the derivatives of tangent, cotangent, secant, and cosecant follow from sine and cosine and the quotient rule?Topic 2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions: derive and apply the derivatives of the remaining trigonometric functions.8 min answer →
- What are the derivatives of the core transcendental functions sine, cosine, the natural exponential, and the natural logarithm?Topic 2.7 Derivatives of cos x, sin x, e to the x, and ln x: state and apply the derivatives of the four basic transcendental functions.8 min answer →
- How are differentiability and continuity related, and where can a continuous function fail to be differentiable?Topic 2.4 Connecting Differentiability and Continuity - Determining When Derivatives Do and Do Not Exist: explain that differentiability implies continuity but not conversely, and identify where derivatives fail to exist.8 min answer →
- How do you estimate a derivative from a table or graph when you do not have a formula?Topic 2.3 Estimating Derivatives of a Function at a Point: estimate the value of a derivative from a table of values or a graph using nearby secant slopes.8 min answer →
- How does the power rule let you differentiate any power of x without going back to the limit definition?Topic 2.5 Applying the Power Rule: differentiate power functions using the power rule, including negative and fractional exponents.8 min answer →
- Why is the derivative of a product not the product of the derivatives, and what rule replaces it?Topic 2.8 The Product Rule: differentiate a product of two functions using the product rule.8 min answer →
- How do you differentiate a quotient of two functions, and why does the order of terms in the numerator matter?Topic 2.9 The Quotient Rule: differentiate a quotient of two functions using the quotient rule.8 min answer →
Unit 3: Differentiation: Composite, Implicit, and Inverse Functions
Module overview →- How do you differentiate a function built by composing one function inside another?Topic 3.1 The Chain Rule: differentiate composite functions using the chain rule.9 min answer →
- If you know how fast a function changes, how fast does its inverse change, without ever finding a formula for the inverse?Topic 3.3 Differentiating Inverse Functions: find the derivative of the inverse of a function at a point using the reciprocal relationship.9 min answer →
- What are the derivatives of arcsin, arctan and the other inverse trigonometric functions, and where do those algebraic expressions come from?Topic 3.4 Differentiating Inverse Trigonometric Functions: state and apply the derivatives of the inverse trigonometric functions.9 min answer →
- What does it mean to differentiate a derivative, and what does the second derivative tell you?Topic 3.6 Calculating Higher-Order Derivatives: find second and higher-order derivatives and interpret their notation.9 min answer →
- How do you find a slope when y is not isolated, but tangled together with x in an equation?Topic 3.2 Implicit Differentiation: find the derivative of a relation defined implicitly by an equation in x and y.9 min answer →
- Faced with a complicated function, how do you decide which differentiation rules to use, and in what order?Topic 3.5 Selecting Procedures for Calculating Derivatives: choose and combine the appropriate differentiation rules for a given function.9 min answer →
Unit 4: Contextual Applications of Differentiation
Module overview →- When a derivative comes from a real situation, what does its value actually mean, and what are its units?Topic 4.1 Interpreting the Meaning of the Derivative in Context: interpret a derivative as a rate of change in an applied setting, with correct units.9 min answer →
- When two changing quantities are linked by an equation, how does the rate of one determine the rate of the other?Topic 4.4 Introduction to Related Rates: relate the rates of change of two quantities connected by an equation through implicit differentiation in time.9 min answer →
- When a limit gives the indeterminate form 0/0 or infinity/infinity, how can derivatives rescue it?Topic 4.7 Using L'Hospital's Rule for Determining Limits of Indeterminate Forms: evaluate limits of indeterminate form using L'Hospital's rule.9 min answer →
- How can a tangent line be used to estimate the value of a function near a known point?Topic 4.6 Approximating Values of a Function Using Local Linearity and Linearization: use the tangent line to approximate function values near a point.10 min answer →
- Beyond motion, how do derivatives describe rates of change in problems about populations, temperature, flow and cost?Topic 4.3 Rates of Change in Applied Contexts Other Than Motion: model and interpret rates of change in non-motion applied settings.9 min answer →
- What is the step-by-step method for solving a full related-rates word problem from start to finish?Topic 4.5 Solving Related Rates Problems: solve complete related-rates problems using a structured method.10 min answer →
- How are position, velocity and acceleration connected, and how do you tell when a moving particle speeds up or changes direction?Topic 4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration: analyze the motion of a particle along a line using derivatives.10 min answer →
Unit 5: Analytical Applications of Differentiation
Module overview →- How are the properties of a function, its first derivative, and its second derivative connected, and how do you justify conclusions about one from another?Topic 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative: justify conclusions about f using f-prime and f-double-prime.9 min answer →
- How does the sign of the second derivative tell you about concavity and points of inflection?Topic 5.6 Determining Concavity of Functions over Their Domains: use the second derivative to find concavity and points of inflection.9 min answer →
- How does the sign of the first derivative tell you where a function is increasing or decreasing?Topic 5.3 Determining Intervals on Which a Function Is Increasing or Decreasing: use the sign of the first derivative to find intervals of increase and decrease.9 min answer →
- How do you find extrema and analyze the behavior of a curve defined implicitly?Topic 5.12 Exploring Behaviors of Implicit Relations: analyze extrema and concavity of implicitly defined relations using implicit differentiation.9 min answer →
- What guarantees a continuous function has a maximum and minimum, and where can extrema occur?Topic 5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points: identify critical points and distinguish local from global extrema.9 min answer →
- How do you turn a real-world maximum or minimum question into a calculus problem?Topic 5.10 Introduction to Optimization Problems: set up an optimization problem by writing the quantity to be optimized as a function of one variable.8 min answer →
- How do you use the first and second derivatives to sketch a function, and read between the graphs of f, f-prime and f-double-prime?Topic 5.8 Sketching Graphs of Functions and Their Derivatives: use derivative information to sketch a graph and relate the graphs of f, f-prime and f-double-prime.9 min answer →
- How do you solve a full optimization problem and justify that you have found the absolute maximum or minimum?Topic 5.11 Solving Optimization Problems: solve a complete optimization problem and justify the absolute extremum.10 min answer →
- How do you find the absolute maximum and minimum of a function on a closed interval?Topic 5.5 Using the Candidates Test to Determine Absolute (Global) Extrema: find absolute extrema by comparing values at critical points and endpoints.8 min answer →
- How does a sign change in the first derivative classify a critical point as a local maximum or minimum?Topic 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema: classify critical points using sign changes in the first derivative.8 min answer →
- When does the Mean Value Theorem guarantee a point where the instantaneous rate equals the average rate, and how do you use it?Topic 5.1 Using the Mean Value Theorem: state the hypotheses and conclusion of the MVT and apply it to find a guaranteed point.9 min answer →
- How does the sign of the second derivative at a critical point classify it as a local maximum or minimum?Topic 5.7 Using the Second Derivative Test to Determine Extrema: classify critical points using the sign of the second derivative.8 min answer →
Unit 6: Integration and Accumulation of Change
Module overview →- What algebraic properties let you split, combine and reverse definite integrals?Topic 6.6 Applying Properties of Definite Integrals: use linearity, additivity over intervals, and limit-reversal properties of definite integrals.8 min answer →
- How do left, right, midpoint and trapezoidal sums approximate the area under a curve, and when do they over- or under-estimate?Topic 6.2 Approximating Areas with Riemann Sums: approximate area using left, right, midpoint, and trapezoidal sums, and reason about over- and under-estimates.10 min answer →
- How do you evaluate an integral with an infinite limit or an unbounded integrand using limits?Topic 6.13 Evaluating Improper Integrals: evaluate integrals with infinite limits of integration or an infinite discontinuity by rewriting them as limits of proper integrals, determining convergence or divergence (BC).10 min answer →
- How does the area under a rate-of-change graph represent the accumulated change in a quantity?Topic 6.1 Exploring Accumulations of Change: interpret the area under a rate graph as the net accumulated change in a quantity.8 min answer →
- How do you find indefinite integrals of basic functions by reversing the differentiation rules?Topic 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation: find indefinite integrals of power, trigonometric, exponential and reciprocal functions.9 min answer →
- How does integration by parts reverse the product rule to integrate a product of functions?Topic 6.11 Integrating Using Integration by Parts: integrate products of functions by reversing the product rule, choosing the parts and applying the formula, including repeated use (BC).10 min answer →
- How does u-substitution reverse the chain rule to integrate composite functions?Topic 6.9 Integrating Using Substitution: integrate composite functions by reversing the chain rule with u-substitution, including changing limits for definite integrals.10 min answer →
- How do you analyze the increasing, decreasing and concavity behavior of an accumulation function from the graph of its integrand?Topic 6.5 Interpreting the Behavior of Accumulation Functions Involving Area: analyze extrema and concavity of an accumulation function using the graph of the integrand.9 min answer →
- How does the limit of a Riemann sum become the definite integral, and what does summation and integral notation mean?Topic 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation: express a Riemann sum in summation notation and define the definite integral as its limit.9 min answer →
- How do you choose the right antidifferentiation technique for a given integral?Topic 6.14 Selecting Techniques for Antidifferentiation: choose between rewriting, basic rules, and substitution to evaluate an integral.9 min answer →
- What is an accumulation function, and how does the Fundamental Theorem of Calculus give its derivative?Topic 6.4 The Fundamental Theorem of Calculus and Accumulation Functions: differentiate accumulation functions using the first part of the Fundamental Theorem of Calculus.9 min answer →
- How does the Fundamental Theorem of Calculus let you evaluate a definite integral using an antiderivative?Topic 6.7 The Fundamental Theorem of Calculus and Definite Integrals: evaluate definite integrals using the second part of the Fundamental Theorem of Calculus.9 min answer →
- How do you integrate a rational function by splitting it into linear partial fractions?Topic 6.12 Using Linear Partial Fractions: rewrite a rational function with distinct linear factors in the denominator as a sum of partial fractions and integrate each to a logarithm (BC).9 min answer →
Unit 7: Differential Equations
Module overview →- How does Euler's method use the slope at each step to approximate a solution curve numerically?Topic 7.5 Approximating Solutions Using Euler's Method: approximate the solution of a differential equation at a point using Euler's method with a given step size and initial condition (BC).10 min answer →
- How does the differential equation for proportional growth give the exponential model, and how do you use it?Topic 7.8 Exponential Models with Differential Equations: derive and apply the exponential growth and decay model from a proportional-rate differential equation.9 min answer →
- How do you solve a separable differential equation for its general solution?Topic 7.6 Finding General Solutions Using Separation of Variables: solve a separable differential equation for the general solution.9 min answer →
- How does an initial condition pin down one particular solution from the general family?Topic 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables: solve an initial value problem by separation and applying the initial condition.10 min answer →
- How does the logistic differential equation model bounded growth toward a carrying capacity?Topic 7.9 Logistic Models with Differential Equations: model and analyze bounded growth with the logistic differential equation, identifying the carrying capacity and the point of fastest growth (BC).10 min answer →
- How do you translate a description of a rate of change into a differential equation?Topic 7.1 Modeling Situations with Differential Equations: write a differential equation from a verbal description of a rate of change.8 min answer →
- How do you use a slope field to estimate solution curves and describe long-term behavior?Topic 7.4 Reasoning Using Slope Fields: sketch solution curves on a slope field and reason about their behavior.8 min answer →
- What is a slope field, and how do you draw one from a differential equation?Topic 7.3 Sketching Slope Fields: construct a slope field by computing the slope at grid points from a differential equation.8 min answer →
- How do you check whether a given function is a solution of a differential equation?Topic 7.2 Verifying Solutions for Differential Equations: verify that a proposed function satisfies a differential equation by substitution.8 min answer →
Unit 8: Applications of Integration
Module overview →- How do you find the length of a smooth curve using a definite integral?Topic 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled: compute the length of a curve y = f(x) and the distance a particle travels using the arc length integral (BC).9 min answer →
- How do you find the area between two curves given as functions of x?Topic 8.4 Finding the Area Between Curves Expressed as Functions of x: integrate the top minus the bottom curve to find the enclosed area.9 min answer →
- When is it easier to find an area by integrating with respect to y, and how do you do it?Topic 8.5 Finding the Area Between Curves Expressed as Functions of y: integrate right minus left with respect to y to find the enclosed area.9 min answer →
- How do you find the area between curves that cross more than once and swap which is on top?Topic 8.6 Finding the Area Between Curves That Intersect at More Than Two Points: split the area at each crossing where the top and bottom curves swap.9 min answer →
- How do you recover velocity and position from acceleration, and find displacement and total distance, using integrals?Topic 8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals: use integrals to find velocity, position, displacement and total distance.10 min answer →
- How do you find the average value of a continuous function over an interval using a definite integral?Topic 8.1 Finding the Average Value of a Function on an Interval: compute the average value of a function with the definite-integral formula.8 min answer →
- How do you use a definite integral of a rate to find the net change in an applied quantity?Topic 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts: find net change in a quantity by integrating its rate in context.10 min answer →
- How do you find the volume of a solid of revolution using the disc method about the x- or y-axis?Topic 8.9 Volume with Disc Method: Revolving Around the x- or y-Axis: find the volume of a solid of revolution about a coordinate axis using the disc method.9 min answer →
- How do you find the volume of a solid of revolution about a horizontal or vertical line other than a coordinate axis using the disc method?Topic 8.10 Volume with Disc Method: Revolving Around Other Axes: find the volume of a solid of revolution about a line other than a coordinate axis using the disc method.9 min answer →
- How do you find the volume of a solid of revolution with a hole using the washer method about the x- or y-axis?Topic 8.11 Volume with Washer Method: Revolving Around the x- or y-Axis: find the volume of a solid of revolution with a hole about a coordinate axis using the washer method.10 min answer →
- How do you find the volume of a solid of revolution with a hole about a line other than a coordinate axis using the washer method?Topic 8.12 Volume with Washer Method: Revolving Around Other Axes: find the volume of a solid of revolution with a hole about a line other than a coordinate axis using the washer method.10 min answer →
- How do you find the volume of a solid with square or rectangular cross sections built on a region?Topic 8.7 Volumes with Cross Sections: Squares and Rectangles: integrate the cross-sectional area to find volume when cross sections are squares or rectangles.9 min answer →
- How do you find the volume of a solid with triangular or semicircular cross sections built on a region?Topic 8.8 Volumes with Cross Sections: Triangles and Semicircles: integrate the cross-sectional area to find volume when cross sections are triangles or semicircles.9 min answer →
Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions
Module overview →- How do you differentiate a curve defined by parametric equations to find its slope?Topic 9.1 Defining and Differentiating Parametric Equations: define a curve parametrically and find dy/dx as the ratio of the parametric derivatives (BC).10 min answer →
- How do you differentiate a vector-valued function to find velocity and acceleration?Topic 9.4 Defining and Differentiating Vector-Valued Functions: define a vector-valued function and differentiate it component by component to find velocity and acceleration (BC).9 min answer →
- How do you describe a curve in polar coordinates and find its slope dy/dx?Topic 9.7 Defining Polar Coordinates and Differentiating in Polar Form: convert between polar and Cartesian coordinates and find dy/dx for a polar curve r = f(theta) (BC).10 min answer →
- How do you find the length of a parametric curve over an interval of the parameter?Topic 9.3 Finding Arc Lengths of Curves Given by Parametric Equations: compute the length of a parametric curve using the integral of the square root of (dx/dt)^2 + (dy/dt)^2 (BC).9 min answer →
- How do you find the area of a region bounded by two polar curves?Topic 9.9 Finding the Area of the Region Bounded by Two Polar Curves: find the area between two polar curves by subtracting one-half r-squared integrals over the correct angle interval, after finding intersections (BC).10 min answer →
- How do you find the area enclosed by a single polar curve using an integral?Topic 9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve: compute the area swept by a polar curve r = f(theta) using the one-half r-squared integral (BC).10 min answer →
- How do you integrate a vector-valued function to recover velocity and position?Topic 9.5 Integrating Vector-Valued Functions: integrate a vector-valued function component by component to recover velocity from acceleration and position from velocity, using initial conditions (BC).9 min answer →
- How do you find the second derivative of a parametric curve to determine concavity?Topic 9.2 Second Derivatives of Parametric Equations: find d^2y/dx^2 for a parametric curve by differentiating dy/dx with respect to t and dividing by dx/dt, and use it for concavity (BC).9 min answer →
- How do you solve planar motion problems using parametric and vector-valued functions?Topic 9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions: find position, velocity, speed, acceleration, displacement and distance for a particle moving in the plane (BC).10 min answer →