Ace Calculus 1 in 9 Hours (The Complete Course)

Review algebra by learning how to simplify equations through factoring. Identify the greatest common factor, apply differences of squares and cubes, and factor quadratics such as x^2+x-4 and 5x^2+7x-6.

Explore how functions map inputs to outputs, graph key types—linear, quadratic, cubic, rational, square root—and identify one-to-one versus many-to-one mappings using a free graphing tool.

Explore how functions are defined by domain and range, using union and intersection of sets, and examine examples like 1/x and sqrt(x) to identify possible inputs and outputs.

Identify vertical and horizontal asymptotes as lines a function never reaches, such as x = 0 for 1/x and y = 0 for e^x.

Explore composition functions, where an input to a function is another function, and learn notation like f∘g with clear examples such as f(x)=√x and g(x)=2x+1.

Learn how inverse functions map outputs back to inputs, determine domain and range, and identify when inverses exist (one-to-one), using swapping variables and solving examples like f(x)=(x+1)/(x+7) and f^{-1} notation.

Explore how to evaluate limits using substitution and algebraic techniques, distinguishing when cheating works from when you must simplify with conjugates or factorization, and identifying indeterminate forms and vertical asymptotes.

Apply the rules of limits to constants, products, quotients, and sums; bring constants outside the limit, and verify right-hand limits while ensuring denominators are not zero.

Master one-sided limits by analyzing piecewise and absolute value functions as x approaches from the left and right. Learn when these limits exist or do not exist through concrete examples.

Explore infinite limits and vertical asymptotes with left and right limit behavior, sign analysis, and substitution techniques using rational functions and specific examples.

Explore limits at infinity and horizontal asymptotes in rational functions, showing how leading terms determine limits as x grows without bound and as x tends to negative infinity.

Master limits of trigonometric functions by reviewing radians, the unit circle, and sine, cosine, and tangent relationships, then apply limit rules and the squeeze theorem.

Apply the squeeze theorem to bound x^2 sin(1/x) between -x^2 and x^2 and deduce the limit is 0 as x approaches 0, highlighting its use with bounded trigonometric functions.

Examine continuity by analyzing left and right limits, the regular limit, and f(a) using plots. Distinguish removable and jump discontinuities, and identify holes and vertical asymptotes.

This lecture analyzes continuity through examples, verifies the three continuity conditions, identifies a vertical asymptote at t=3, confirms continuity at t=4, and examines a piecewise function not continuous at x=-3.

Explore polynomials and polynomial functions, learn how to identify leading terms and degrees, and analyze continuity across piecewise definitions, including left and right limits at key points.

Explore left-continuity and right-continuity, define left and right limits, and apply continuity criteria at a using f(x)=x^2 on [0,2], where the right limit at 2 fails.

Explore left and right limits on a graph, identify vertical and horizontal asymptotes, determine when regular limits exist, and assess continuity at key points.

Explains left and right continuity and continuity on a closed interval. A function is continuous at points in the interval, right continuous at a, and left continuous at b.

Illustrate the intermediate value theorem: a continuous function on [a, b] with opposite endpoint signs has a root in (a, b), demonstrated with a polynomial root between 1 and 2.

Explore secant lines as average rate of change and relate them to tangent lines as instantaneous rate of change, using limits to show the slope converges.

Define the derivative as the slope of the tangent line via the limit, with f(x)=x^2 giving f'(x)=2x and g(x)=√x giving g'(x)=1/(2√x), and find the tangent to x^3 at x=2.

Master the power rule for derivatives of x^n with n a positive integer, where the derivative is n x^(n-1). Recognize that constant functions have zero derivatives.

Explore derivative notation for functions, including f'(x), y', dy/dx, and x^p forms; apply the power rule and recognize that constants have zero derivative, with prime and Leibniz notations being interchangeable.

Master the constant multiple rule by differentiating c·f(x) as c·f'(x), applying the power rule, and checking with examples like 6x^(-5) and 6x^2.

Learn the sum and difference rules: derivative of a sum is the sum of derivatives, with y = 3x^2 + 4x, using the constant multiple rule and the power rule.

Learn how to differentiate products using the product rule, where the derivative equals F'G + FG'. Observe the rule in action with examples like x^3·x^2 and a limit-based proof.

Learn to apply the quotient rule to differentiate f(x)/g(x), using (f'(x) g(x) − f(x) g'(x)) / [g(x)]^2, with examples like x^7/x^4 and (x+1)/(x−1).

Examines differentiability, defining derivative existence via limits, with examples like the absolute value function at zero and x cubed, and proves differentiability implies continuity, while continuity may not imply differentiability.

Explore the normal line, perpendicular to the tangent line, and learn that their slopes multiply to -1. Using f(x)=4x^4+x+1 at x=1, determine the normal line equation.

Explore higher order derivatives from first to sixth derivative for a polynomial, using both prime notation and Leibniz notation, and see how higher derivatives eventually reach zero.

Master the chain rule for derivatives of composed functions by identifying the inside and outer functions, using Lebanese notation, and applying related rules with practical examples.

Master implicit differentiation for equations that define Y implicitly, using circle examples and the chain and product rules to find Y'.

Apply implicit differentiation to x^3 + y^3 = x + 1 at (0,1) to get y' = (1 - 3x^2)/(3y^2) and the tangent line y = (1/3)x + 1.

Review trigonometric functions on the circle, defining sine, cosine, tangent, secant, cosecant, and cotangent, and establish core identities including sin^2+cos^2=1 and 1+tan^2=sec^2.

Derivatives of trig functions are derived using limits and rules, yielding sin' x = cos x and cos' x = -sin x, with product, quotient, and chain rule applications.

Explore derivatives of trigonometric functions through sine of cosine, applying the chain rule to a cube-root expression, and practice implicit differentiation with step-by-step techniques.

Explore inverse trig functions by restricting domains to make sine, cosine, and tangent one-to-one, then derive their inverses as arc sine, arc cosine, and arc tangent.

Explore derivatives of inverse trigonometric functions using implicit differentiation and triangle reasoning. Apply product and chain rules to arcsin and arctan examples, deriving standard formulas and practice problems.

Review exponential and logarithmic rules, including properties of exponents, log identities, change of base, Euler's number e, and basic graph behaviors of exponential and logarithmic functions.

Differentiate exponential and log functions, prove d/dx e^x = e^x, apply product and chain rules, and derive log base b of x using change of base and 1/x.

Use log differentiation to differentiate functions with nonconstant base and exponent, such as B^X and X^X, by taking logs and applying implicit differentiation.

Explore several examples of log differentiation for functions with x-dependent base and exponent, using implicit differentiation and bringing the exponent in front of the log to find y'.

Revisit limits and indeterminate forms, applying L'Hôpital's rule to 0/0 and infinity over infinity cases, supplementing with algebraic simplification and derivatives to find key limits.

Explore indeterminate limits such as infinity minus infinity and zero times infinity. Apply tricks like common denominators, L'Hôpital's rule, and exponential forms to evaluate limits such as X^X and (1+1/X)^X.

Explore how derivatives express rates of change and the slope of a tangent, using x squared to show average and instantaneous rates, plus velocity and acceleration from displacement.

Explore rectilinear motion through three examples, computing average and instantaneous velocity from displacement functions, then determine speed, acceleration, and maximum height for moving objects.

Derive the first and third kinematic equations from displacement, velocity, and acceleration using calculus, showing how initial velocity and initial position drive displacement over time.

Explore differential equations, learn how to verify solutions and distinguish between general and particular solutions, with examples and boundary and initial conditions, and connect to simple harmonic motion.

Derive the simple harmonic motion differential equation from Hooke's law and Newton's second law, express the general solution as A cos(Ω t) + B sin(Ω t), and define amplitude, period, and frequency.

Derive the simple harmonic motion of a 2 kg mass on a spring with displacement 0.2 m and zero initial velocity, yielding x(t) = 0.2 cos(8 t).

Apply the mean value theorem to a function continuous on [a,b] and differentiable on (a,b) to guarantee a c with f'(c) equal to the secant slope. Use f(x)=x^3-3x+2 on [-2,2].

Understand Rolle's theorem as a special case of the mean value theorem. If the derivative is zero on an interval, the function is constant, and endpoints equality guarantees zero-derivative point.

Use derivatives to estimate values with differentials and linear approximation by tangent lines, showing how f(a+Δx) ≈ f(a) + f'(a) Δx and Δy = f'(a) Δx.

Use differentials and linear approximation to estimate values near known points, applying tangent-line concepts to problems like sqrt(36.1) and cylinder volume with small radius changes.