AP Calculus AB

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.

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 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.

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.

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.

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.

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.

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}.

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 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.

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.

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.