AP Calculus AB

Explore derivatives as the instantaneous rate of change and the slope of the tangent line, using the limit definition f'(x) = lim_{h->0} (f(x+h)-f(x))/h with examples.

Explore how derivatives correspond to slopes via tangent lines on graphs, and learn where differentiability fails—sharp turns, discontinuities, vertical tangents, and cusps—using the limit definition.

Explore where a function fails to be differentiable, including sharp turns, discontinuities, cusps, and vertical tangents, and distinguish differentiability from continuity through key examples and the derivative definition.

Present a practical tour of basic derivative rules, including the power rule, constant and constant multiple rules, and the sum rule, with tangent line and trig function applications.

Apply basic derivative rules to locate horizontal tangents by setting f'(x)=0, solving polynomials and trig cases, and finish by solving a tangent condition with f'(x)=g'(x) to find k.

Master basic derivative rules while learning continuity and differentiability through practice problems, including solving for constants and derivatives, and analyzing a diver’s motion for instantaneous and average velocity.

Master product and quotient rules to differentiate products and quotients, and learn trig derivatives for tangent, secant, and cosecant using u and v methods.

Practice applying product and quotient rules to compute derivatives from function values and graph slopes, using f and g and their derivatives. Find horizontal tangents by solving f'(x)=0.

Master the chain rule for derivatives of composite functions by identifying inner and outer parts, differentiating the outer, and multiplying by the inner derivative, with trig, power, and quotient examples.

apply the chain rule and quotient rule to composite functions, simplify to remove negative exponents, and derive first and second derivatives to locate tangent lines and horizontal tangents.

Explore inverse functions and the derivative of inverses, using the formula f inverse prime of a equals one over f prime of f inverse of a, with worked examples.

learn how to differentiate exponential functions, such as y = e^{3x^2}, using product and quotient rules, algebraic simplification, and find tangent lines and horizontal slopes.

Learn the natural logarithm properties, including ln1=0, ln e =1, and ln(ab)=ln a+ln b; apply rules to simplify expressions and differentiate using d/dx ln(u)=u'/u.

Learn to differentiate bases other than e using log base a rules and logarithmic differentiation, applying implicit differentiation to functions like (x+1)^(x-3).

Learn inverse trig functions and their derivatives, including arcsin, arctan, and arccos, with memorized derivative rules and chain-rule practice, plus tangent line applications.

Apply the chain rule to implicit functions and solve for dy/dx using algebra. Practice with examples that mix x and y, including product rules and tangent line calculations.

Practice decoding implicit differentiation using product and quotient rules to find dy/dx and d^2y/dx^2, while plugging in a point and keeping results in terms of x and y.

Master related rates using implicit differentiation, differentiate area and functions with respect to time, relate radius and height, and compute dA/dt and dh/dt in circle and cone problems.

Master L'Hôpital's rule to evaluate indeterminate limits by differentiating the top and the bottom, applying it repeatedly to cases like 0/0 or infinity/infinity, with quick examples.

Use the first derivative to identify critical numbers, determine intervals where f' is positive or negative, and locate relative maximums and minimums for increasing and decreasing functions.

Identify intervals of increasing and decreasing using the first derivative test, locate critical numbers where f'(x)=0, and determine relative maxima and minima.

Apply the second derivative test to identify concavity and inflection; with f'' positive, concave up, and with f'' negative, concave down; use f' signs for increasing, decreasing, maximums, minimums.

Learn to find relative maxima and minima with the second derivative test: identify critical points, classify concavity (f''>0 or <0), and handle f''=0 by the first derivative test, with examples.

Apply optimization by setting the derivative to zero and checking sign changes to minimize time, and minimize metal in can design using volume and surface area.

Apply linear approximation via the tangent line and derivative. Use f(x)=x^2 near 1 to estimate f(1.01), then cube root of 127 near 125 to estimate 5.0267 with small error.

Learn to estimate the area under a curve using left, right, midpoint Riemann sums and the trapezoid rule, illustrated with speed-time graphs to compute distance.

Apply left and right Riemann sums and the trapezoid rule to estimate area under a curve for a tank scenario. Add the initial 150 gallons to the accumulated amount.

Learn how to use Riemann sums to approximate area and how the definite integral gives exact area under a curve, using limits, delta x, and splitting the integral at C.

Explore antiderivatives and indefinite integrals, learn reverse differentiation, add constants of integration, and solve differential equations using initial conditions to obtain particular solutions.

Apply the fundamental theorem of calculus to solve initial-value problems by evaluating definite integrals from a to b, find the particular solution from initial conditions, and determine Y3.

Apply the fundamental theorem of calculus to determine F(8) from F(3)=5 and the integral from three to eight equals 20, then break and evaluate piecewise definite integrals.

Estimate gallons after 24 hours with a midpoint Riemann sum using three subintervals and trapezoid rule, starting from 150 gallons, and apply the integral to find the average water flow.

Explain the second fundamental theorem of calculus—the derivative of an integral with a variable upper limit—and show how g(x) = ∫0^x f(t) dt reveals monotonicity, concavity, and key extrema.

Master integration techniques for exponential functions in AP Calculus AB, using u-substitution, chain rule reversal, and shortcuts to integrate expressions like e^{3x+1}, x e^{-x^2}, and e^{7x}.

Master definite integrals with u-substitution, changing limits, and integrating exponential and trigonometric functions. Solve differential equations and apply rate problems in the context of exponential intervals.

Explore how to compute the area between two curves using top minus bottom, or right minus left for y-variables, with setup and crossing points.

Learn to find the area between curves using definite integrals, locate intersections with a calculator, and set up multi-part integrals for regions bounded by curves.

Compute the area of the region between y = 3x and y = x^2 from x = 0 to 3, then determine the volume from equilateral triangle cross sections.

Review the area between two curves and volumes using disks and washers (not shells), set bounds from intersections, and rotate about y = -3 or x = 5 with cross-sections.

Learn to analyze particle motion with derivatives and integrals: derive velocity from position, find rest points, compute displacement, distance, and determine speeding up or slowing down from velocity and acceleration.

Analyze one-dimensional particle motion using position, velocity, and acceleration to determine when the particle moves right or left, rests, and speeds up or slows down with calculus and calculator graphing.

Learn to solve differential equations by separating variables, handling implicit derivatives, integrating, and using initial conditions to obtain the particular solution y as a function of x.

Derive differential equations solutions using separation of variables, natural logs and exponentials, apply initial conditions, and analyze a wolf population model with a particular solution and domain constraints.

Use separation of variables to solve differential equations for exponential growth and decay, determine k from initial data, and apply to bacterial growth and bear population models.

Apply Newton's law of cooling, a differential equation, to model temperature change. Solve with separation of variables and natural logs, then analyze a sqrt(P) growth model for population.

Learn to draw slope fields for AP calculus AB, where dy/dx depends on x, on y, and on both; sketch solution curves through given points and note weather forecasting connections.