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.

Apply derivatives using data to estimate rates of water flow, including average rate of change and inverse of w, using calculators for f'(15) and from 4 to 20.

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.

Apply the chain rule to find second derivatives, handle constants, and evaluate at specific values. Practice composing derivatives with f and g and apply to harmonic motion and velocity.

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.

Explore implicit differentiation to find dy/dx, derive the tangent line, and identify vertical tangents by solving 3y^2+2y-5=0, yielding four coordinates on the curve.

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.

Study related rates through shadow problems and trough volume questions, using similar triangles and volume formulas to derive dx/dt and dh/dt.

Explore the mean value theorem, showing how a function's derivative matches its average rate of change at some point, with visual intuition and a speeding-car application.

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.

Identify absolute and relative extrema on intervals by using derivatives to find critical points, then apply the extreme value theorem and evaluate the original function at endpoints.

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.

Learn to analyze f using derivatives f' and f'': locate where f is increasing or decreasing, identify relative maxima and minima, and determine concavity and inflection points from f' behavior.

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.

Explore optimization of square and circle shapes by minimizing area under a fixed perimeter, and determine a cost-minimizing box given volume and material costs.

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.

Master integration by u-substitution, a reverse chain rule. Choose u as the inner expression, compute du, substitute, and then re-substitute with the original variable.

Apply u-substitution to integrate a range of expressions, from cube-root forms like (x^2+4)^(1/3) to trig powers, by matching inner derivatives and rewriting steps for clean integration.

Learn the fundamental theorem of calculus, evaluate definite integrals using antiderivatives, and compute area under curves with limits, with examples and calculator strategies.

Learn how to perform u-substitution with definite integrals, convert limits from x to u, and evaluate examples involving area under curves and related applications.

Learn how to handle tricky u-substitution cases by solving for x and using a double u-substitution, adjusting limits, and applying even-odd symmetry to simplify definite integrals.

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.

Explore the average value of a continuous function using the mean value theorem and the one over (B minus A) integral formula, with perfect-rectangle intuition and examples like speed.

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 the natural log function and its integration, covering the rule ∫1/x = ln|x|, definite integrals, u-substitution, and long division with examples.

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.

Learn to set up volumes of solids with known cross sections by integrating top minus bottom or right minus left, using rectangle, semicircle, and equilateral triangle cross sections.

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.

Learn to compute volumes using the disk and washer methods by revolving regions about the x-axis and y-axis, applying top-minus-bottom radii and pi r^2 integrals.

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.

Develop differential equations skills using separation of variables, natural logs, and exponential solutions, applying initial conditions. Solve AP-style problems, including a weight-growth model and puppy weight example.

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.