A Rapid Introduction To Calculus

Explain core sets in calculus, including natural numbers, integers, rationals, and real numbers, their subset relationships, and interval types such as open and closed, plus unions, intersections, and Venn diagrams.

Explore quadratic functions and quadratic equations in standard form with a nonzero leading coefficient. Use the discriminant delta to determine root counts: two, one, or undefined.

Learn to solve inequalities by converting to equality, identifying zeros and points of undefinedness, and testing subranges to determine sign in rational expressions, through worked examples.

Explore the absolute value concept, its piecewise definition, and how it measures distance between numbers, while applying product and quotient properties and the triangle inequality for calculus foundations.

Explore the natural (maximal) domain and the range of functions with examples, including x squared and radical and rational expressions, showing how to identify values that keep expressions defined.

Learn what injections (one-to-one functions) are, prove f(x1)=f(x2) implies x1=x2 for a nonzero slope, and see monotone functions as injections with examples.

Define surjections as functions whose image equals the entire target set, and show with f(x)=2x+1 from R to R that the image is all real numbers. Solve f(x)=b to find x in the domain with f(x)=b, proving onto.

Explore function composition by applying g from A to B and f from B to C, showing how f∘g yields a new C value and that g∘f is not commutative.

Learn how to determine bounded functions and prove that f(x) equals 1 over x squared plus one is bounded by 1 on its domain, using the definition of bounds.

Define elementary functions as those formed by addition, subtraction, multiplication, division, and composition. Include polynomials, rational functions, logarithms, exponentials, and trigonometric functions in calculus.

Explore one sided limits as x tends to a from the left and right, and the theorem shows the limit exists only when both sides converge to the same value.

Explore continuity and discontinuity of functions by examining definitions, open and closed intervals, one-sided limits, and classic discontinuities, then review how sums, differences, products, and quotients preserve continuity.

Explore the intermediate value theorem for continuous functions on a closed interval, which guarantees a root when f(a) and f(b) have different signs, and apply it to prove a polynomial has at least two roots on [-2, 2].

Motivate integration by approximating the area under a nonnegative curve on [a,b] with rectangles, then define the definite integral and the indefinite integral as antiderivatives.

Compute basic antiderivatives by splitting complex expressions into simple parts, applying rules for x^n, 1/x, and x, using ln|x| with an indefinite integral plus c, and verify with differentiation.

Master integration by substitution using the chain rule, setting u = g(x), and transforming integrals such as ∫ e^{x^3} 3x^2 dx into ∫ e^{u} du.

Practice solving an integral by using u-substitution with u = 1/x, recognizing 1/x^2 and the derivative of 1/x, and rewriting the integrand as 1 plus 1/x or 1/x plus 1.