Calculus 1 Mastered

Trace the progression from natural numbers and integers to rational and irrational numbers, then real, imaginary, and complex numbers, with their symbols and subset relationships.

Master graphing with this mouse, a versatile calculator offering online and offline modes, derivatives, integrals, inequalities, and parametric equations, with live editing and labeled points.

Explore the definition of a function, learn how to evaluate it, and review domain, range, exponential, and logarithmic functions as foundational topics for limits in calculus i.

Define functions as a machine that maps each input (domain) to exactly one output (range) using a rule like f(x)=x^2, ensuring unique associations.

Evaluate a function by substituting numbers into placeholders, applying the rule, and identify independent and dependent variables, with domain and range as inputs and outputs.

Explore the domain of a function as the set of all inputs that keep the function defined, with real numbers as inputs and examples showing when outputs become undefined.

Identify the range of a function by contrasting it with the domain, treat functions as machines, and determine that f(x) = x^2 produces nonnegative outputs.

Learn to identify one-to-one functions, defined as functions with unique outputs for each input, using the horizontal line test and examples like x^2 versus x+1.

Explore how inverse functions work for one-to-one functions and learn to find them with three-step method: set y = f(x); solve for x; swap x and y to obtain inverse.

Explore exponential functions defined by f(x)=a^x, with a>0 and a≠1, where any real base is raised to a real power, and apply examples.

Explore the natural exponential function with base e, the irrational number approx 2.71828. Understand f(x)=e^x and how e relates to limits like (1+1/n)^n, and see basic graph behavior.

Describe how logarithmic functions invert exponential functions, using a base and exponent and properties like log base a of 1 equals 0.

Explore natural logarithms, their key properties and domains, including ln(1)=0, ln e=1, ln(e^x)=x, and domain rules for expressions like ln(x+3) and ln(3-x).

Explore the core logarithm laws, including product, quotient, and power rules, and learn the change-of-base formula with practical examples.

Introduce trigonometric ratios for a right triangle, define the hypotenuse, opposite, and adjacent, and explain sine, cosine, tangent along with cosecant, secant, and cotangent.

Use a point on the terminal side to form a right triangle and derive sine, cosine, and tangent from x, y, r, with quadrant signs.

Explore how inverse trigonometric functions are defined by restricting domains to obtain one-to-one functions, and learn how arc sine, arc cosine, and arc tangent specify principal ranges.

Explore the concept of limits, from informal to precise definitions, including one-sided limits and the laws of limits. Learn to apply these ideas to solving problems.

Explore the limit concept by examining how a function behaves as x approaches a value from the right-hand side and the left-hand side, and determine existence when these limits agree.

Determine whether a limit exists by comparing left and right values as x approaches a point, using step function and oscillating graph examples.

Learn how one-sided limits define the behavior of a function as x approaches a, using right-hand and left-hand limits and examples like step functions.

The limit laws lecture introduces the basic rules for limits, including sum, difference, constant multiple, product, quotient, and powers, plus substitution and evaluation with examples.

Apply limit laws to evaluate various limits, including products, quotients, powers, and constants. Explore concrete examples, such as limits as x approaches two or minus two, to reinforce techniques.

Explore limit techniques for rational functions, including factoring and cancellation, and the conjugate method to handle square roots, to solve diverse limit problems.

Explore the squeeze (sandwich) theorem and its use when limit laws fail, illustrated with cos x/(sin x−2) and x^2 cos(1/x) to bound and determine limits.

Apply the squeeze theorem to evaluate limits. Bound x^2 sin(1/x) by -x^2 and x^2 to get limit 0, and bound x cos(3/x) - 5 to get limit -5.

Learn the precise epsilon-delta definition of a limit, detailing how for every epsilon there exists a delta that constrains x near c so that |f(x)-L|<epsilon.

Students learn the precise definition of limit through delta-epsilon proofs, identifying f(x), l, c, and deriving delta from epsilon, illustrated with three examples.

Explore limits at infinity by examining how functions behave as x grows toward positive and negative infinity. Use leading terms and degrees in rational functions to determine these limits.

Evaluate limits at infinity by comparing degrees and leading coefficients, and by dividing by the highest power to identify dominant terms. Apply these rules to rational and square root expressions.

Explore how limits at infinity determine horizontal and oblique asymptotes for rational functions, including when degrees match or differ, and how leading coefficients and division yield the asymptote equations.

Explore infinite limits and vertical asymptotes by examining how f(x) behaves near a point, demonstrating right and left limits approaching positive or negative infinity.

Explore the definition and conditions for continuity, examine the properties of continuous functions, and study limits and the intermediate value theorem for continuous functions.

Define continuity by ensuring the limit of f(x) as x approaches c exists and equals f(c), confirming the three conditions for a continuous function.

Explore the types of discontinuity: removable, jump, infinite (asymptotic), and essential, and how limits determine continuity on intervals.

Identify continuity and discontinuities by determining the domain and applying standard results for polynomials, rational, root, and trigonometric functions.

Explore how continuous functions preserve continuity under addition, subtraction, multiplication, and division, and analyze complex expressions by decomposing into continuous pieces while considering domain.

Explains the intermediate value theorem for a continuous function on a closed interval, showing that if f(a) and f(b) straddle L, a c exists with f(c) = L.

Explore the fundamentals of derivatives by examining rate of change and average rate of change, then learn the definition and rules of derivatives to differentiate functions, with attached practice downloads.

Explore rate of change and average rate of change, comparing slopes over intervals, and introduce instantaneous rate of change as the basis for derivatives.

Explore instantaneous rate of change as the limit of the average rate of change, yielding the derivative f'(x1) and the tangent line slope at a point.

Explore the derivative definition as the instantaneous rate of change via the limit as h approaches zero, yielding the tangent slope. Learn notations f'(x), dy/dx, and D/dx.

Apply the derivative definition by computing f(x+h) and taking the limit as h approaches zero. The examples include differentiating a polynomial, a constant, and a rational function (x+1)/(x-1).

Identify why certain points are non-differentiable, such as cusps, sharp turns, fast oscillation, breaks, or undefined values, where the tangent line or slope fails to exist.

Master the constant rule, derivative of a constant is zero, and the power rule, bring the exponent down and subtract one.

Apply the constant multiple rule to differentiate c·f(x) by leaving the constant untouched and using the power rule on f(x), illustrated with 5x^4, 2x, and 5x^-5.

Differentiate using the sum and difference rule by differentiating each term and preserving the signs; e.g., 3x^2 + 5x - 1 becomes 6x + 5.

Master the product rule for differentiating h(x)=f(x)g(x) by taking f'(x)g(x) + f(x)g'(x), and extend to three or more factors.

Master the quotient rule for differentiating a function as a quotient, h(x)=f(x)/g(x). Compute h'(x) = (f'(x)g(x) - f(x)g'(x)) / [g(x)]^2, and apply it in examples with f and g.

Master the chain rule for differentiating composite functions, identify outer and inner functions, and apply derivative rules with practical examples and an online calculator for practice.

Learn to differentiate composite expressions and quotients by simplifying the inner function and applying chain, quotient, and power rules to square root problems, with worked examples.

Explore derivative symbols and notations, including f'(x), dy/dx, and the derivative operator d/dx, and learn how different mathematicians used varied forms to express rates of change.

Learn to sketch the graph of a function's derivative by identifying zeros, sign patterns, and whether the slope is increasing or decreasing, with practice examples.

Explore higher order derivatives, from the first to the second, third, and beyond, learn the notation f', f'', and f^(n), and see how they relate to concavity and graphing.

Learn to find the equation of the tangent line to a function at a point using the derivative, and apply the formula with examples.

Learn how to differentiate trig functions, including sine, cosine, tangent, secant, cosecant, and cotangent, with rules and worked examples using product and quotient rules.

Master derivatives of trigonometric functions using the chain rule, learning to differentiate sin(kx), cos(kx), tan(kx), and powers like sin^2 x, with inner-angle derivatives.

Learn how to differentiate inverse trig functions using rules for arcsin, arccos, arctan, arcsec, and arccsc, applying the chain rule and standard differentiation formulas.

Explain and apply derivatives of inverse trigonometric functions using quotient, product, and chain rules through multiple examples, including arcsin and arccos cases.

Explore implicit differentiation by treating y as a function of x, apply the chain rule, and solve for dy/dx across implicit equations.

Learn the derivative of inverse functions using the rule (f inverse of x)' = 1 / f'(f inverse of x); ensure one-to-one and apply to examples.

Discover how to differentiate the natural exponential function, e^x, and its general form e^u using the chain rule, with practical examples.

Apply the rule d/dx ln(u) = (1/u) du/dx for u > 0; transform and differentiate forms like ln(x^2+1) and sqrt(ln x) and ln(6/x^2) to reveal easy derivatives.

Learn to differentiate general exponential functions a^u(x) with a>0, using d/dx a^u = a^u ln a · u'(x). Apply it to 2^x, 7^(2−5x+4), and 5^(sin(2x)).

Master the derivative of logarithmic functions, including base a, using d/dx log_a x = 1/(x ln a). Apply with examples like log_2(5x) and log_6(x^2-5x+4).

Learn how logarithmic differentiation simplifies derivatives of complex functions by taking natural logs on both sides and using log rules, with x^x yielding derivative x^x(ln x + 1).

Explore the applications of derivatives, including related rates and optimization, and learn how to extract information from derivatives to analyze and graph functions.

Master related rates by following five steps: draw a picture, name variables, note givens and goals, differentiate with respect to time, and relate to dV/dt and dr/dt.

Explore related rates with examples: differentiate circle area to find radius change, and use Pythagorean relation to find rate of distance between car and truck.

Use related rates in a conical tank: relate radius and height by similar triangles, differentiate the cone volume, and obtain dy/dt = 25/(9π) m/min at y = 3 m.

Apply the Pythagorean theorem to a ladder problem and differentiate to find how fast the top slides down, how the triangle area changes, and the rate of change of theta.

Apply derivatives to optimization by outlining the problem, drawing a picture, defining variables, and testing critical and end points to maximize the open-top box volume.

We minimize the cylinder's surface area subject to a fixed volume of one liter, deriving r = (500/pi)^(1/3) and h = 2r for the least material cost.

Maximize the window area by optimizing the semicircular top and rectangular bottom with 12 meters of framing; use derivatives to express area in radius and height and locate the maximum.

Explore the concepts of absolute maximum and minimum (global extrema) and local maximum and minimum (local extrema). Identify absolute and local extremes on a function's graph, using neighborhoods and intervals.

Explore critical points, where the derivative is zero or undefined, identify global and local extrema among interior points, and use the first derivative test to find maxima and minima.

Use the first derivative test to locate critical points where the derivative is zero or undefined, then identify local or global minimum and maximum by checking increasing or decreasing behavior.

Identify the critical points of f(x)=2x^5-x^4-2x^3 by solving f'(x)=0, reveal stationary point at x=0 and local extrema at x=-3/5 (local max) and x=1 (local min); apply first derivative test.

Learn how the second derivative reveals a function's concavity, distinguishing concave up from concave down, and identify inflection points as the switch between these behaviors.

Use the second derivative to determine concavity and inflection points. Apply testing around second-derivative zero or undefined values to quadratic and more complex examples.

Apply the second derivative test to critical points to identify local extrema. If f''<0, it's a local max; if f''>0, it's a local min; if f''=0, the test fails.

Learn a five-step method to graph functions using domain, symmetries, asymptotes, intercepts, and first/second derivatives, illustrated by f(x)=x^4-2x^2 and its concavity and inflection points.

Master graphing a rational function using five steps: determine domain, asymptotes, intercepts, and symmetry; analyze derivatives for critical points and concavity to sketch the graph.

Master L'Hôpital's rule to evaluate limits using derivatives, converting 0/0 or ∞/∞ forms into derivative ratios, and repeat if needed. Practice with sin x over x and (x^2-1)/(x^2+3x-4).

Turn indeterminate forms into solvable quotients with the lobster rule, solving zero over zero, infinity over infinity, indeterminate product, difference, and power cases.

Learn Rolle's theorem as a special case of the mean value theorem. It requires continuity on [a,b], differentiability on (a,b), and equal endpoints, ensuring a c with f'(c)=0.

Master the mean value theorem: continuous on [a,b], differentiable on (a,b), guarantees a c where the average rate of change equals the instantaneous rate; Rolle's theorem is a special case.

Apply the mean value theorem to a 200-kilometer trip, showing that an average speed of 100 km/h implies a moment when speed equals 100 km/h, explaining a speeding ticket.

Achieve mastery of calculus one by mastering limits with the precise definition, solving problems on continuity, derivatives, and applications of derivatives.