AP Calculus AB: The 18-Hour Fast Track to a Perfect Score

Explore algebra review techniques to simplify equations, factor out greatest common factors, and apply differences of squares, differences of cubes, sum of cubes, and quadratic factoring.

Map inputs to outputs using functions and distinguish one to one from many to one. Graph quadratic, linear, cubic, square root, and rational functions with desmos, noting coordinates and asymptotes.

Define a function by its domain and range, with union and intersection. For example, 1/x has domain all real numbers except zero, and sqrt(x) has domain x≥0 and range y≥0.

Identify horizontal and vertical asymptotes, including x=0 for 1/x and y=0. Note e^x stays positive with y=0 as a horizontal asymptote, and learn these via limits; next, composition functions.

Learn how composition functions combine two functions by substituting one function into another. See examples using f(x)=x^2 and g(x)=x^2−1, and express f∘g and g∘f with sqrt(2x+1).

Explore inverse functions by swapping x and y and solving for y, determine domain and range, and identify when a function is one-to-one, with examples.

this lecture introduces limits using f(x) = (x^2-4)/(x-2), shows how values approach four as x approaches two from either side, though f(2) is undefined, and simplifies to x+2.

Substitute the input to evaluate limits when possible, then use algebraic tricks such as difference of squares and rationalization for indeterminate forms. Identify cases where the limit does not exist.

Master limit rules: constant limits, sums, differences, products, and quotients via distributive property. Apply them to evaluate limits like (2 f(x))/(h(x) - g(x)) as x approaches zero, yielding -6/5.

Explain one-sided limits using a piecewise function; show left-sided limit at 3 is 5 and right-sided limit is 2, so the limit does not exist, plus absolute value examples.

Analyze infinite limits and vertical asymptotes by studying f(x)/g(x) as denominator tends to zero, where limits do not exist or approach positive or negative infinity, noting left and right behavior.

Explore limits at infinity and horizontal asymptotes, showing how f(x)=1/x tends to zero, and how factoring out the leading term in rational expressions identifies asymptotic lines y=0, y=5/2, and y=2.

Explore limits of trigonometric functions by reviewing radians, unit circle, sine, cosine, tangent, and cosecant, secant, cotangent; relate trig functions and apply limit rules, including the squeeze theorem.

The squeeze theorem: if f ≤ g ≤ h near a and lim f = lim h = L, then lim g = L.

Explore continuity by examining left and right limits, regular limits, and function definition at a point, including removable and jump discontinuities and the three point definition test.

Explore continuity with concrete examples, verify the conditions, note a vertical asymptote at t = 3, continuity at t = 4, and piecewise continuity at 0 but not at -3.

Explore polynomials, their degree and leading term, and how polynomial functions stay continuous everywhere. Analyze a piecewise function to show continuity at x=1 but not at x=3.

Explore left and right continuity by analyzing left and right limits and f(a). In f(x)=x^2 on [0,2], the right limit fails at 2, while left continuity holds.

Examine how limits and continuity appear on graphs by evaluating left and right limits, identifying vertical and horizontal asymptotes, and noting f(x) values at specific points.

Explore left and right continuity and determine when f(x) is continuous on a closed interval, noting right continuity at a and left continuity at b, with jumps or undefined points.

apply the intermediate value theorem to a continuous function on [a,b], ensuring a c with f(c)=n between f(a) and f(b); illustrated by a root between 1 and 2.

Explore secant lines as the average rate of change between two points, and tangent lines as the limit of secant slopes, illustrating the instantaneous rate of change with x^2 examples.

Define the derivative as the tangent slope via a limit. Derive f'(x)=2x for x^2, g'(x)=1/(2 square root of x); and compute tangent to x^3 at 2 with slope 12 and intercept -16.

Apply the power rule by showing that f(x)=x^n differentiates to n x^(n-1) for positive integers, and note constants have zero derivative and how to rewrite expressions to use the rule.

The lecture compares derivative notations, including prime and Leibniz forms like dy/dx and df/dx, and shows the power rule, constant function derivatives, and the zero derivative for constants.

Master the constant multiple rule to differentiate functions like y = 6/x^5 and y = sqrt(5x) by pulling constants out and applying the power rule, with proofs via limits.

Apply the sum and difference rules to derivatives, using constant multiple and power rules on polynomials like 3x^2+4x^5, and prove the derivative of a sum equals the sum of derivatives.

Master the product rule for derivatives, using f g' plus f' g to differentiate products. See examples with x^3 times x^2 and a limit-based proof of the rule.

Master the quotient rule for derivatives, understanding that d/dx [f(x)/g(x)] = (f'(x) g(x) - f(x) g'(x)) / [g(x)]^2, with counterexamples and a step-by-step proof.

Explore differentiability by examining when derivatives exist, including absolute value, vertical tangents, and discontinuities, and learn that differentiability implies continuity, but not vice versa.

Determine the normal line by the negative reciprocal of the tangent slope from the derivative; for f(x)=4x^4+x+1 at x=1, the normal line is y = -1/17 x + 103/17.

Explore higher order derivatives, from the second to the sixth derivative, using classic and Leibniz notation, with a polynomial example where higher derivatives eventually become zero.

Learn the chain rule for derivatives of composition functions, with f(x)=x^3 and g(x)=x^2+1, and apply dy/dx = f'(g(x))·g'(x).

Learn how implicit differentiation handles equations that implicitly define y as a function of x, using the circle example and the chain rule to find dy/dx.

Use implicit differentiation on x^3 + y^3 = x + 1 to find dy/dx, then evaluate at (0,1) for slope 1/3 and tangent line y = (1/3)x + 1.

Review sin, cos, tan identities and their squares: sin^2+cos^2=1, sec^2=1+tan^2, csc^2=1+cot^2, from the circle, and prepare for derivatives of transcendental functions.

Derive derivatives of sine, cosine, and tangent using the limit definition with product, chain, and quotient rules, and extend to secant, cosecant, and cotangent.

Apply the chain rule to derivatives of trigonometric functions, including sine of cosine x and the cube root of secant x plus five, and practice implicit differentiation for dy/dx.

Learn how restricting trig functions to 1-to-1 domains enables inverses like arcsin, arccos, and arctan, with defined ranges and unit circle references.

Derive formulas for inverse trigonometric functions using implicit differentiation and the chain rule, with right-triangle relationships for sine and cosine, and the derivatives of inverse sine, cosine, and tangent.

Prepare for derivatives by reviewing key exponential and logarithmic rules, including powers, products, quotients, and change of base, plus natural log concepts.

Explore derivatives of exponential and logarithmic functions, including d/dx e^x, product and chain rules, implicit differentiation, and the change of base formula for log base b of x.

Revisit limits and master indeterminate forms using L'Hôpital's rule and chain rule. Explore zero over zero and infinity over infinity with practical examples to compute limits.

Master indeterminate forms in ap calculus ab, such as infinity minus infinity and zero over zero, using algebra tricks and L'Hôpital's rule, apply exponential and logarithmic techniques to evaluate limits.

Explore how derivatives measure rate of change, from the tangent slope to instantaneous rate, with the x^2 example linking average rate of change to velocity and acceleration.

Explore three rectilinear motion examples, deriving average and instantaneous velocity from f(t) and s(t), then analyze height models to find maximum height, velocity at impact, and acceleration.

Learn to derive the kinematic equations from a displacement function using calculus, via derivatives and integrals, linking displacement, velocity, and acceleration to time with initial velocity and position.

Explore differential equations, linking derivatives to functions, and verify solutions. Learn general and particular solutions, boundary and initial conditions, with examples including y' + 3x^2 y = 6x^2.

Derive the simple harmonic motion differential equation from Hooke's law and Newton's second law, arriving at x'' + ω^2 x = 0, its general solution, amplitude, period, and frequency.

Analyze a 2 kg mass on a spring; compute k=128 and omega=8, then solve for x(t) with x(0)=0.2, v(0)=0, giving x(t)=0.2 cos(8 t).

Learn to use derivatives to estimate nearby values via differentials and linear approximation, where f(a+dx) ≈ f(a) + f'(a) dx, the tangent line giving the change in y.

Apply linear approximation and differentials to estimate values using tangent lines, from sqrt(36.1) to cylinder volume changes with radius.

Apply the mean value theorem to continuous and differentiable functions on [a,b], showing a c where f'(c) equals the secant slope; for example, f(x)=x^3-3x+2 on [-2,2] yields c=±2/√3.

Rolle's theorem states that if f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then there exists c in (a,b) with f'(c)=0, a special case of mean value theorem.

Showcases testing curve orthogonality by finding intersections, differentiating to get slopes, and confirming perpendicular tangents via product equals -1, using y = x^2 and x^2 + 2y^2 = 1.

Identify where a function increases or decreases by examining derivative signs (positive or negative) on intervals and applying the increasing/decreasing test, illustrated with the example f(x)=x^3+3x^2-9x+4.

Local maxima and minima occur at critical points where f' is zero or undefined. An absolute maximum occurs at P, with no absolute minimum since graph extends to negative infinity.

Identify critical points where f'(x)=0 or f' is undefined and f is defined, as shown by several examples including x^3-3x^2-72x+15 yielding x = -3 and x = 4.

Use the first derivative test to locate local maxima and minima by analyzing the derivative's sign with a sine table, finding -3 as a maximum and 4 as a minimum.

Explore concavity and inflection points by testing the second derivative: f''>0 for concave up and f''<0 for concave down, with x=1 as the inflection in the example.

Use the second derivative test to identify local extrema for critical points where f'(c)=0: f''(c)>0 yields a local min, f''(c)<0 yields a local max, f''(c)=0 requires the first derivative test.

The extreme value theorem states that a continuous function on [a,b] has an absolute maximum and minimum. Evaluate f at critical points and endpoints to locate them.

Follow eight steps to sketch a curve: determine domain and vertical and horizontal asymptotes, intercepts, and use a first and second derivative sign table to locate maximum and inflection points.

Sketch a cubic function using first and second derivative tests to locate the local max at x=1, inflection at x=2, and local min at x=3, with y-intercept -4.

Sketch the function f(x)=2x^2/(x^2-1) by locating the domain, vertical asymptotes at ±1, horizontal asymptote y=2, and intercepts, then analyze derivatives to map increasing, decreasing, and concavity.

Formulate the objective as a function of x and y, apply first and second derivative tests with extreme value theorem, and optimize sphere, cylinder, and cone volumes and surface areas.

Three optimization problems: maximize rectangle area with fixed perimeter, minimize open-top box surface area at fixed volume, and locate the nearest point on a parabola.

Learn area estimation via Riemann sums to approximate area under a curve from a to b. Use rectangles with width delta x and heights from endpoints or midpoints, then sum.

Estimate the area under sin(πx) on [0, 2] using a six-rectangle left-endpoint Riemann sum. See how signed area can cancel, illustrating Riemann sums as signed area.

Estimate displacement using a six-rectangle right-hand Riemann sum for v(t) on 0 to 3, illustrating the signed area equals displacement and distance traveled sums positive magnitudes.

Discover sigma notation, the language for summing a_k terms. See how to expand and translate between expanded form and sigma notation, from k=1 to n and from k=0 to 25.

Master summation rules and formulas, including the constant multiple and addition rules, and memorize key sums like n, n(n+1)/2, n(n+1)(2n+1)/6, and [n(n+1)/2]^2 for area of a function.

Master evaluation of summations with sigma notation by applying the four standard sums, using constant multiples, and rewriting index ranges to compute examples with k, k^2, and k^3.

Compute the exact signed area under a continuous function on [a, b] using a Riemann sum with right side points, delta x, and take the limit as n approaches infinity.

Compute the signed area under f(x)=x−x^2 on [0,3] using a Riemann-sum approach, derive x_k, substitute, and take the limit to obtain the exact area, 9.

Evaluate signed area of the function 1 minus half x on 0 to 4 using a sum of rectangles, noting the limit is zero as positive and negative areas cancel.

Compute the signed area from 0 to 2 using delta x and f(x_k) in a Riemann sum. Derive the general form A(0 to r) = r^3 + r^2 + r.

Use the four-step method to compute the signed area from a to b. Derive the result b^2 - b - a^2 - a, matching the initial guess.

Explore the fundamental theorem of calculus part II, linking antiderivatives and definite integrals with notation ∫_a^b f(x) dx, and work through examples like polynomials, e^x, and arctan.

Explore basic antiderivative formulas using power rules, exponentials, and trig and inverse trig integrals, illustrated with practical examples.

Prove the fundamental theorem of calculus part II by applying the mean value theorem to show that the definite integral equals the antiderivative difference, linking f and F.

Explore the properties of the definite integral, including integrability via finite signed area, constant multiple, additivity, symmetry (odd and even functions), and comparisons.

examine three definite integrals: zero from -pi to pi due to odd symmetry; bound sin(x^2) by one on 0 to pi/3; split 0 to 4 and apply the fundamental theorem.

Explain part i of the fundamental theorem of calculus: if f is continuous on [a, x], then d/dx ∫_a^x f(t) dt = f(x), showing the signed area depends on x.

Prove the fundamental theorem of calculus, part i, by showing that if f is continuous and F is an antiderivative, then d/dx ∫_a^x f(t) dt = f(x).

Explore how to differentiate and evaluate integrals using the fundamental theorem of calculus, variable bounds, even/odd symmetry, and the chain rule.

Learn the rules of antiderivatives and the constant of integration, using initial conditions to determine c. Find F(x) for x^2+e^x with F(0)=4, giving F(x)=x^3/3+e^x+3.

Apply antiderivatives to physics problems by deriving velocity and displacement from acceleration, using initial conditions, and compare no-air-resistance and air-resistance models with exponential terms.

Explore solving a non-basic antiderivative using substitution, algebraic methods, and long division, with emphasis on expansion, distribution, and splitting the numerator.

Use substitution for non-basic antiderivatives by identifying the inner function in a composition and confirming u' appears in the integrand, then substitute and back-substitute.

Master u-substitution by replacing inner functions with u and du to turn integrals into basic antiderivatives. Explore examples with trigonometric, rational, and exponential forms, and adjust bounds for definite integrals.

Learn to apply the algebraic method to non-basic antiderivatives by expanding, splitting the numerator, and using u-substitution, with examples converting to basic antiderivatives and handling composition functions.

Explore three modified basic rules for antiderivatives, using u-sub, to yield arc sine, arc tangent, and arc secant forms, with examples using a=3 and absolute value cases.

Master the completion of the square to convert challenging integrals into standard forms, using substitution and resulting in arctan, arcsin, and arcsec solutions as shown in examples.

Use long division for improper rational functions when the numerator degree exceeds the denominator; obtain a quotient and remainder to integrate, with substitutions such as u=x^2+4.

Introduce differential equations, define independent and dependent variables, and distinguish verifying solutions from solving them; verify and solve examples like y''+4y=0 and y'-3y=0.

Learn to solve separable differential equations by separating variables, integrating both sides, and solving for y as a function of x using a constant determined by initial conditions.

Explore separable differential equations through two examples: derive x(t) from velocity by integration with an initial condition, and solve for velocity under air resistance using Newton's second law.

Explore solving separable differential equations by separating variables, integrating both sides, and applying initial conditions to determine constants, with ellipse and Newton's cooling examples.