A Rapid Introduction To Calculus

Define a set as a collection of objects, ignoring repetitions and order, using squiggly braces; explain membership, the set of all real x such that p(x), and subset relations.

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.

Explore key algebraic identities, including (a+b)^2 = a^2+2ab+b^2, (a−b)^2 = a^2−2ab+b^2, and a^2−b^2 = (a−b)(a+b), with examples and limit applications.

Factor trinomials by finding two numbers that multiply to a times c and sum to b, rewrite the quadratic, and factor to (3x+2)(x-2).

Learn to solve inequalities by converting them to equalities, locating zeros and undefined points, and testing subintervals, with graphing and interval notation to verify where expressions are positive or negative.

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.

Define a function as a relation from a domain to a range with a unique output for every input. Compare functions to nonfunctions and hint at bijections and monotone functions.

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.

Determine the natural domain of the function by ensuring x(3-x) is positive due to the square root and nonzero due to the denominator, yielding the interval (0,3).

Explore monotone functions on an interval, distinguishing monotone and strictly monotone increasing or decreasing, with a formal proof using the definition, and a real-line example f(x) = 4 - 3x.

Prove that f(x) = x^2 − 2x is increasing on x ≥ 1 by showing f(x1) < f(x2) whenever x1 < x2, using algebraic rearrangements and positivity of x1, x2.

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 the image as the subset of the range produced by the domain. For f(x)=x^2, the image is [0,∞), illustrating nonnegative values, a constant case, and surjection.

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 inverse functions exist for bijections, visualize the inverse as a reflection across line y = x, and apply this to linear functions like f(x)=ax+b with inverse g(x)=(x−b)/a.

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.

Discover even and odd functions, with evenness defined by f(-x)=f(x) and oddness by f(-x)=-f(x). Using f(x)=x^4+5x^2-3, confirm its evenness and show symmetry about the y-axis.

Explore polynomials defined by coefficients and degree, with a leading coefficient determining the highest power, and rational functions as polynomials divided by polynomials, with P(x)=0 having at most n solutions.

Explore exponents with y = a^x, where a > 0 and a ≠ 1. Recognize logarithms as the inverse, and use log_a(xy) = log_a x + log_a y.

Explore trigonometric functions by defining sine and cosine on the unit circle, relate them to radians and 2 pi, explain periodicity and parity, and analyze tangent's domain and vertical asymptotes.

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

Learn the concept of a limit by substituting values near a, explore right and left neighborhoods, and illustrate with x^2 and (x^2-9)/(x-3) to show substitution, continuity, and cancellation.

Examine several limit examples, using substitution for defined points and algebraic factorization to cancel terms. Apply the difference of squares trick for square-root limits and recognize continuous cases.

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 infinite limits by examining how 1/x^2 diverges near zero, 1/x diverges to infinity from the right and negative infinity from the left, and how x^α behaves as x grows.

Apply limit arithmetic to sums, differences, products, and quotients, including infinity limits, and identify indeterminate forms such as infinity minus infinity, infinity times zero, or zero divided by zero.

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].

The lecture explains that every odd-degree polynomial has at least one root, and shows with a cubic example how limits, continuity on real line, and intermediate value theorem guarantee it.

Explore the motivation for differentiable functions by linking velocity, slope, and the tangent, and relate two derivative definitions using x0 and h with a limit as h tends to zero.

Define differentiable functions by a limit and derivative notation, with the example f(x)=x^2+2x. Compute the derivative of ax+b using the second definition and note the local, continuous nature of differentiability.

Explore a continuous but non-differentiable example using the absolute value function |x| at zero, demonstrating continuity via one-sided limits and non-differentiability at zero.

Compute the right and left limits of the absolute value function at zero to show the derivative does not exist. This illustrates that differentiability is stronger than continuity.

Not every elementary function is differentiable. For example, the absolute value function is elementary but not differentiable at zero.

Demonstrate that if a function is differentiable at x0, then it is continuous at x0 by using the derivative definition and limit rules to show lim_{x→x0} f(x) = f(x0).

Compare derivative notations, including Lagrange's prime and Leibniz dy/dx, with occasional reference to Newton's dot, and illustrate using x^2 to show forms for future integrals.

Explore standard derivatives: constants; power rules (x^n -> n x^{n-1}); e^x -> e^x; a^x -> a^x ln a; ln x -> 1/x; and sin, cos, tan derivatives with sec^2 x.

Learn the rules of differentiation, including constant factor, sum, difference, product, quotient, and chain rules, with examples like derivatives of x^2 and 2^x.

Learn how to compute the derivative of an inverse function using the chain rule, proving that (f^{-1})'(y0) = 1 / f'(x0) when f is invertible and f'(x0) is nonzero.

derive the derivative of log base a of x using the inverse relation to a^x and the chain rule, yielding d/dx log_a x = 1/(x ln a).

Derive the derivative of the inverse of x^n, i.e., the nth root, and obtain (x^{1/n})' = (1/n) x^{1/n - 1}.

If a function is differentiable, continuous, and monotone with nonzero derivative near a point, it has an inverse that is monotone, and the inverse’s sign matches the function’s derivative.

Define f(x) = ln(1+x) - (x - x^2) and prove f'(x) > 0 for x>0, then f(x) > f(0) = 0, giving x - x^2 < ln(1+x) for all x>0.

Explore fermat's theorem in calculus, linking local extrema to zero derivatives in differentiable functions and showing how boundaries and non-differentiable or stationary points determine absolute maxima and minima.

Rolle's theorem asserts that for a function continuous on a closed interval [a,b] and differentiable on (a,b) with f(a)=f(b), there exists c in (a,b) where f'(c)=0.

Lagrange's theorem asserts that a function continuous on [a, b] and differentiable on (a, b) has c with f'(c) equals the slope (f(b) - f(a)) / (b - a).

View Rolle's theorem as a special case of Lagrange's theorem and prove it from Lagrange using continuity on [a,b], differentiability on (a,b), and f(a)=f(b), yielding f'(c)=0 for c in (a,b).

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.

Define antiderivatives as functions whose derivative equal s f(x) and relate them to the definite integral. illustrate with f(x)=x^2, F(x)=x^3/3 and G(x)=2x^3/3+5, and note that all antiderivatives differ by a constant.

Define the indefinite integral as the family of antiderivatives whose derivative is the integrand, with a constant c. Illustrate with common integrals like ∫2x dx equals x^2 + c.

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.

tackle an integration exercise by recognizing the linear inner function 2x-7, applying substitution u=2x-7, and using the power rule to obtain 6 times (2x-7) to the two-thirds power plus C.

Apply integration by parts to a polynomial times the natural logarithm, compare the f and g' options, and derive the antiderivative.

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.

Use a u-substitution with u = 1/x to simplify the integral, convert dx/x^2 to -du, and complete the antiderivative to obtain 1 + 1/x + c.