The fundamental theorem of calculus, part 2 says that if f is continuous on [a, b] and F is any antiderivative of f, then the integral of f from a to b equals F(b) − F(a). To evaluate a definite integral, find an antiderivative, plug in the top and bottom limits, and subtract.
The fundamental theorem of calculus, part 1 says that if f is continuous, the accumulation function F(x) = ∫ from a to x of f(t) dt has derivative F′(x) = f(x). The rate at which area builds up under a curve equals the height of the curve.
The properties of definite integrals are rules that let you split, combine and rearrange integrals without computing them from scratch. The main ones: integrals over adjacent intervals add; reversing the limits changes the sign; constants factor out; and the integral of a sum is the sum of the integrals.
A definite integral, written ∫ from a to b of f(x) dx, is the signed area between the graph of f and the x-axis from x = a to x = b. It is defined as the limit of Riemann sums as the number of rectangles goes to infinity. Area above the x-axis counts as positive, and area below it counts as negative.
The trapezoidal rule estimates the area under a curve with trapezoids instead of rectangles. With n equal pieces of width Δx, the estimate is Tₙ = (Δx/2)[f(x₀) + 2f(x₁) + … + 2f(xₙ₋₁) + f(xₙ)]. It equals the average of the left and right Riemann sums and is usually more accurate than either.
A Riemann sum estimates the area under a curve by adding up the areas of rectangles. Split [a, b] into n equal pieces of width Δx = (b − a)/n, make a rectangle on each piece whose height is the function’s value at the left endpoint, right endpoint or midpoint, and add the areas: width × height for each rectangle.
An antiderivative of f is a function F whose derivative is f: F′(x) = f(x). The indefinite integral ∫f(x) dx = F(x) + C stands for all the antiderivatives at once, where C is any constant. For example, ∫2x dx = x² + C, because the derivative of x² + C is 2x.
Newton’s method approximates a zero of a function, a solution of f(x) = 0, by following tangent lines. Start with a guess x₀. Then repeat the formula x(n+1) = x(n) − f(x(n))/f′(x(n)). Each new value is where the tangent line at the old value crosses the x-axis, and the values usually close in on the zero very quickly.
Optimization problems use derivatives to find the largest or smallest value of a real quantity, such as the greatest area, the least cost or the shortest distance. Write the quantity as a function of one variable, find its domain, find the critical points, and check which one gives the maximum or minimum.
Curve sketching means drawing an accurate graph of a function from its formula, using f, f′ and f″ together. From f you get the domain, intercepts, symmetry and asymptotes. From f′ you get where the graph rises and falls and its local extrema. From f″ you get concavity and inflection points.