AP Calculus BC — Free Practice Questions, Study Guides & Mock Exams

Advanced calculus including parametric, polar, vector functions, and series

Topics covered

• Applications of Integration
• Parametric Equations
• Polar Coordinates
• Vectors
• Infinite Sequences and Series
• Taylor and Maclaurin Series
• True/False
• MCQ
• Word Bank
• Trap Classification
• numeric
• Rule Naming
• Parsons
• Fill in the Blank

Free AP Calculus BC study guides

• Introducing Calculus: Can Change Occur at an Instant? — An average rate needs an interval. Shrink the interval to nothing and the limit answers the question.
• Defining Limits and Using Limit Notation — What the function approaches — read from both sides, and independent of the value at the point.
• Estimating Limit Values from Graphs — Trace each branch and read where it is heading. The plotted point is a separate question.
• Estimating Limit Values from Tables — Close in from both sides. Agreement is evidence — and evidence is not proof.
• Determining Limits Using Algebraic Properties of Limits — Split the expression, take each limit, reassemble — but only when every piece exists.
• Determining Limits Using Algebraic Manipulation — A 0/0 form means a factor cancels. Find it, rewrite, and substitute again.
• Selecting Procedures for Determining Limits — No new technique here — only the skill of picking the right one from the shape of the expression.
• Determining Limits Using the Squeeze Theorem — Trap the function between two curves that meet. If they meet, it has nowhere else to go.
• Connecting Multiple Representations of Limits — Picture, table, formula, sentence — one limit, and all four must agree.
• Exploring Types of Discontinuities — Four kinds, told apart by one question: does the two-sided limit exist?
• Defining Continuity at a Point — One equation, three separate conditions — and a complete answer names all three.
• Confirming Continuity over an Interval — Every interior point, plus one side at each endpoint — and every seam checked separately.
• Removing Discontinuities — You can fill a hole, but you cannot bridge a jump. The limit decides which one you have.
• Connecting Infinite Limits and Vertical Asymptotes — A zero denominator is a question, not an answer. Cancel first, then check the sign from each side.
• Connecting Limits at Infinity and Horizontal Asymptotes — End behaviour is a race between numerator and denominator — and you must run it in both directions.
• Working with the Intermediate Value Theorem (IVT) — Three hypotheses, one conclusion, and a justification the reader expects word for word.
• Defining Average and Instantaneous Rates of Change at a Point — One needs an interval and gives a secant slope; the other needs a point and gives a tangent slope.
• Defining the Derivative of a Function and Using Derivative Notation — A secant slope with the gap shrunk to nothing — written five different ways.
• Estimating Derivatives of a Function at a Point — No formula means no limit. Take the best secant the data allows — and say which one you took.
• Connecting Differentiability and Continuity — Four ways a derivative fails — and three of them happen at perfectly continuous points.
• Applying the Power Rule — Bring the exponent down, reduce it by one — once you have rewritten roots and reciprocals as powers.
• Derivative Rules: Constant, Sum, Difference, and Constant Multiple — Differentiation is linear — which is exactly why it does not distribute over a product.
• Derivatives of cos x, sin x, e^x, and ln x — Four results to quote from memory — and one minus sign that decides most of the marks.
• The Product Rule — Differentiate one factor at a time, leave the other alone, and add.
• The Quotient Rule — Bottom times derivative of top, minus top times derivative of bottom, over bottom squared — in that order.
• Finding the Derivatives of Tangent, Cotangent, Secant, and Cosecant — Two results to memorise; the other two are their mirrors with a minus sign.
• The Chain Rule — Differentiate the outside, leave the inside alone, then multiply by the derivative of the inside.
• Implicit Differentiation — Never solve for y. Differentiate both sides and let every y carry its chain-rule factor.
• Differentiating Inverse Functions — Reflection reciprocates slopes — the only trick is finding which point to reflect from.
• Differentiating Inverse Trigonometric Functions — Six results, three to memorise — and every one of them is algebraic.
• Selecting Procedures for Calculating Derivatives — No new rules. The skill is naming the outermost operation before you differentiate anything.
• Calculating Higher-Order Derivatives — Differentiate again. The technique is unchanged — the notation and the meaning are what shift.
• Interpreting the Meaning of the Derivative in Context — Four things in every answer: what is changing, which direction, how fast with units, and when.
• Straight-Line Motion: Connecting Position, Velocity, and Acceleration — One function and two derivatives. The marks are in the signs, and in telling speed from velocity.
• Rates of Change in Applied Contexts Other Than Motion — In minus out. The sign of the net rate answers most of the question before any arithmetic.
• Introduction to Related Rates — Differentiate the relation with respect to time, and every varying quantity brings its own rate.
• Solving Related Rates Problems — The calculus is one chain rule. The marks are in the set-up, the substitution order and the units.
• Approximating Values of a Function Using Local Linearity and Linearization — Use the tangent line in place of the curve — and let concavity tell you which way you erred.
• Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms — Two forms qualify directly. Everything else must be reshaped first — and some things never qualify at all.
• Using the Mean Value Theorem — Somewhere inside the interval, the instantaneous rate equals the average rate — provided both hypotheses hold.
• Extreme Value Theorem, Global Versus Local Extrema, and Critical Points — Closed and bounded guarantees the extremes exist. Critical points and endpoints are where you look for them.
• Determining Intervals on Which a Function Is Increasing or Decreasing — The sign of f' decides everything. Its size decides nothing.
• Using the First Derivative Test to Determine Relative (Local) Extrema — Not where f' is zero — where f' changes sign.
• Using the Candidates Test to Determine Absolute (Global) Extrema — Build a finite list — critical points plus endpoints — evaluate, and pick the extremes.
• Determining Concavity of Functions over Their Domains — Which way the curve bends, decided by f'' alone — and independent of whether it rises or falls.
• Using the Second Derivative Test to Determine Extrema — At a horizontal tangent, the bending decides. One substitution replaces a sign chart — when it works.
• Sketching Graphs of Functions and Their Derivatives — Collect the facts first. The curve that fits them is essentially unique.
• Connecting a Function, Its First Derivative, and Its Second Derivative — Move fluently between the graph, the table, the formula and the sentence — they all describe the same behaviour.
• Introduction to Optimization Problems — Two sentences in the prompt: one names what you optimise, the other states what limits you.
• Solving Optimization Problems — Objective, constraint, one variable, justify — the calculus is the short part.
• Exploring Behaviors of Implicit Relations — The derivative contains y — and that is exactly what distinguishes one branch from another.
• Exploring Accumulations of Change — The area under a rate graph is the amount that accumulated — and area below the axis takes some back.
• Approximating Areas with Riemann Sums — Same slices every time — the only choice is where you sample the height, and which way that pushes the error.
• Riemann Sums, Summation Notation, and Definite Integral Notation — Every symbol in the integral replaces one piece of the sum. Read them side by side and the notation explains itself.
• The Fundamental Theorem of Calculus and Accumulation Functions — Differentiating an integral hands the integrand back — as long as you respect the upper limit.
• Interpreting the Behavior of Accumulation Functions Involving Area — Read the sign of f for direction, its slope for concavity, and its area for value.
• Applying Properties of Definite Integrals — Split, glue and factor without evaluating anything — and know exactly where the properties stop.
• The Fundamental Theorem of Calculus and Definite Integrals — Infinitely many rectangles collapse to two evaluations and a subtraction — provided the function is continuous.
• Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation — Every derivative rule read backwards — plus a constant, and one genuinely exceptional exponent.
• Integrating Using Substitution — The chain rule in reverse: find the inner function whose derivative is already there.
• Integrating Functions Using Long Division and Completing the Square — Neither is an integration technique. Both are algebra that turns an unfamiliar integrand into a familiar one.
• Integrating Using Integration by Parts — The product rule backwards. It trades one integral for another, so choosing u is the whole skill.
• Integrating Using Linear Partial Fractions — Split one hard fraction into two easy ones — each of which integrates to a logarithm.
• Evaluating Improper Integrals — Replace the bad endpoint with a variable, integrate, then take the limit — in that order.
• Selecting Techniques for Antidifferentiation — The hard part is no longer executing a technique — it is choosing one. Classify by structure.
• Modeling Situations with Differential Equations — A differential equation is a sentence about a rate. Translate the English — do not solve it yet.
• Verifying Solutions for Differential Equations — Differentiate the candidate, simplify each side on its own, and see whether they agree everywhere.
• Sketching Slope Fields — Plot the equation itself: at each point, a short tick whose slope is what the equation says it is.
• Reasoning Using Slope Fields — Increasing, concave, asymptotic — all read straight off the ticks, with nothing solved.
• Approximating Solutions Using Euler's Method — Follow the tangent line for one step, recompute the slope, repeat.
• Finding General Solutions Using Separation of Variables — All the y's on one side, all the x's on the other, then integrate both at once.
• Finding Particular Solutions Using Initial Conditions and Separation of Variables — One point selects one curve — and also decides the branch and the domain.
• Exponential Models with Differential Equations — One sentence — rate proportional to amount — and the whole model follows.
• Logistic Models with Differential Equations — Exponential growth with a brake: the rate fades as the population nears its ceiling.
• Finding the Average Value of a Function on an Interval — Sum becomes an integral, count becomes the interval's length. That is the whole formula.
• Connecting Position, Velocity, and Acceleration of Functions Using Integrals — Integrating climbs back up the chain — but it can never recover where the motion started.
• Using Accumulation Functions and Definite Integrals in Applied Contexts — The integral of a rate is an amount — and the exam wants that said in a full sentence.
• Finding the Area Between Curves Expressed as Functions of x — One vertical slice, height top minus bottom, width dx. The integral does the rest.
• Finding the Area Between Curves Expressed as Functions of y — Rotate the whole method a quarter turn: horizontal slices, right minus left, dy.
• Finding the Area Between Curves That Intersect at More Than Two Points — Every crossing starts a new region — and integrating straight through one cancels it away.
• Volumes with Cross Sections: Squares and Rectangles — The base supplies one number per slice. The named shape turns it into an area.
• Volumes with Cross Sections: Triangles and Semicircles — Same integral every time. Draw the slice, name what s measures, then pick the area formula.
• Volume with Disc Method: Revolving Around the x- or y-Axis — Each slice sweeps a coin of radius R. Stack the coins and you have the solid.
• Volume with Disc Method: Revolving Around Other Axes — Same coin, same integral. Only the measurement of R changes — by one subtraction.
• Volume with Washer Method: Revolving Around the x- or y-Axis — The gap between the region and the axis becomes the hole in every slice.
• Volume with Washer Method: Revolving Around Other Axes — Two radii, one shift, applied to both. That is the entire difference.
• The Arc Length of a Smooth, Planar Curve and Distance Traveled — Pythagoras on an infinitesimal step, integrated from one end of the curve to the other.
• Defining and Differentiating Parametric Equations — Both coordinates answer to t, so the slope is the ratio of the two rates.
• Second Derivatives of Parametric Equations — Differentiate the slope with respect to t — then divide by dx/dt one more time.
• Finding Arc Lengths of Curves Given by Parametric Equations — Pythagoras on the two coordinate rates — the integrand is simply the speed.
• Defining and Differentiating Vector-Valued Functions — One function, two components, and differentiation that acts on each independently.
• Integrating Vector-Valued Functions — Two antiderivatives, two constants — and one initial condition that settles both.
• Solving Motion Problems Using Parametric and Vector-Valued Functions — Vector, scalar, or time — decide what kind of answer the question wants before computing.
• Defining Polar Coordinates and Differentiating in Polar Form — r is a distance, not a height — and dr/dθ is not the slope of anything in the plane.
• Finding the Area of a Polar Region or the Area Bounded by a Single Polar Curve — Area is swept, not stacked: thin sectors of radius r and angle dθ.
• Finding the Area of the Region Bounded by Two Polar Curves — Outer sector minus inner sector — with the intersection angles found before anything else.
• Defining Convergent and Divergent Infinite Series — A series is the limit of its partial sums — nothing is ever added infinitely many times.
• Working with Geometric Series — The one family whose exact sum is always available — if the ratio is smaller than one.
• The nth Term Test for Divergence — One limit, one direction: it can prove divergence and never convergence.
• Integral Test for Convergence — Rectangles against area: the sum and the improper integral share a fate, never a value.
• Harmonic Series and p-Series — The ruler every other test measures against — and the boundary at p = 1 is sharp.
• Comparison Tests for Convergence — Swap an awkward series for a familiar one — by inequality, or just by behaviour.
• Alternating Series Test for Convergence — Shrink steadily and reach zero, and the alternating signs do the rest.
• Ratio Test for Convergence — The default whenever factorials or n-th powers appear — and useless the moment L=1.
• Determining Absolute or Conditional Convergence — Strip the signs and test that series first — a convergent answer there settles everything.
• Alternating Series Error Bound — Truncating costs less than the very next term — and its sign tells you which way you erred.
• Finding Taylor Polynomial Approximations of Functions — Match the value and the derivatives at one point: derivatives on top, factorials underneath.
• Lagrange Error Bound — One more term of the polynomial, with the unknown derivative replaced by an upper bound.
• Radius and Interval of Convergence of Power Series — The ratio test settles the interior in one line; the endpoints each need their own.
• Finding Taylor or Maclaurin Series for a Function — Almost never from scratch: substitute into a series you already know.
• Representing Functions as Power Series — The geometric series is a factory: force the shape, substitute, then differentiate or integrate.

Units

• Unit 1
• Unit 2
• Unit 3
• Unit 4
• Unit 5
• Unit 6
• Unit 7
• Unit 8
• Unit 9
• Unit 10

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