Ace Calculus 3 in 16 Hours (The Complete Course)

Explore infinite sequences and infinite series, define a sequence, distinguish convergent from divergent sequences, and identify increasing, decreasing, and bounded properties.

Define sequences as ordered lists with nth term a_n, bracket notation, and infinity dots. Present examples, n/(n+1), sqrt(n-3), Fibonacci, and (-1)^n, and cover convergence via limits and graphs.

Defines convergence of a sequence via its limit L and distinguishes finite limits from divergence. Relates sequence limits to continuous functions, limit rules, and the squeeze theorem with graph examples.

Explore eight examples of sequence convergence and divergence, determining limits such as 1/2^n to 0 and 1/2, with notes on divergence to infinity. Preview monotonic and bounded sequences.

Explore monotonic and bounded sequences, prove when they converge, and examine examples: decreasing sequences like 3/(n+5), n/(n^2+1), and an increasing, bounded recurrence converging to six.

Discover infinite series, compare convergent and divergent sums, and study geometric and p-series, plus seven tests—divergence, integral, direct comparison, limit comparison, alternating, ratio, and root tests.

Explore the definition of a series as an infinite sum, demonstrate convergence with examples like pi/4 via an alternating odd term series, and learn about partial sums and sigma notation.

Explore geometric series, derive the sum a/(1-r) when |r|<1, and identify divergence cases. Apply to examples to determine convergence, finite sums, and when r=1 or |r|≥1.

Explore telescoping series and pairwise cancellation of terms, using partial fractions and log properties across three examples—1/(n(n+1)), 2/(n^2-1), and ln n/(n+1)—to find limits of partial sums.

Explore the harmonic series, the infinite sum of 1/n, and see how regrouping terms proves its divergence. Observe how partial sums grow without bound, preparing the convergence tests to follow.

Apply the divergence test to decide if a series converges or diverges, noting that limit of a_n must be zero for convergence, while the converse may fail; sum n^2/(5n^2+4) diverges.

Use the integral test to compare a series with its improper integral via a positive, continuous, decreasing f(x). See that 1/(n^2+1) converges while 1/n diverges, and p>1 converges.

Estimate the sum of a series convergent by the integral test using partial sums, remainder bounds, and improper integrals; apply to 1/n^3 and determine term counts.

Apply the direct comparison test to determine convergence or divergence by comparing a_n to b_n. Use bounds and p or geometric benchmarks, illustrated with cos^2(n)/(1+n^2), 5/(2n^2+4n+3), and ln k/k.

Use the limit comparison test for positive-term series by ensuring lim a_n/b_n = c with 0 < c < infinity; if one converges or diverges, the other does too.

Explain the alternating test for convergence, requiring decreasing absolute terms and a zero limit; illustrate with the alternating harmonic series, a divergent case, and another convergent using monotone sequence theorem.

Estimate the sum and error of an alternating series using the alternating series estimation theorem. Demonstrate |r_n| ≤ b_{n+1} with partial sums and factorial example, yielding s ≈ 0.368 (n=6).

Differentiate absolute and conditional convergence for alternating series and show that absolute convergence implies convergence, with examples: (-1)^{n-1}/n^2 is absolutely convergent and cos(n)/n^2 is also absolutely convergent.

Compute limit of |a_{n+1}/a_n| to apply the ratio test: if it is <1, the series converges absolutely (and converges if alternating); if >1 or infinity, it diverges; if =1, inconclusive.

Use the root test by evaluating the limit of the nth root of |a_n|; if the limit is <1, the series is absolutely convergent; if >1, it diverges; if =1, inconclusive.

Summarizes seven convergence tests, including geometric and p-series, and outlines strategies for testing series using direct and limit comparisons, alternating, ratio, and root tests.

Explore power series as a function of x, learn convergence and interval of convergence, and master three methods to find power series representations, building on infinite series from previous sections.

Explore the power series definition, its centered form and radius of convergence, and determine the interval of convergence using the ratio test and endpoint checks.

Explore radius of convergence and interval of convergence for power series, use the ratio test on two examples, and verify endpoints to determine final convergence intervals.

Learn how to express functions as power series, using geometric series, differentiation, and integration; explore the radius of convergence, endpoint tests, and applications like integration and differential equations.

Learn to express functions as Taylor and Maclaurin series, find coefficients from derivatives at a, and apply to determine radius of convergence, with e^x as a classic example.

Verify a function has a power series representation by showing the Taylor remainder tends to zero, using Taylor's inequality and three methods: direct computation, differentiation, and power series operations.

Apply the direct computation method to express functions as Maclaurin and Taylor series, derive sine x, explore center shifts, binomial series, and radius of convergence.

Apply term-by-term differentiation and integration of known power series to obtain Maclaurin series; derive cosine from sine and arctangent, and use binomial series to approximate inverse hyperbolic sine to x^5.

Learn two methods to obtain Maclaurin and Taylor series: direct derivatives and term-by-term differentiation or integration, then memorize key series including e^x, sin x, cos x, arctan x, ln(1+x).

Learn to find Taylor series using summation, multiplication, and division of power series, applying Maclaurin series for e^x, e^{-x}, to derive cosh x, e^x sin x, and tan x.

Apply Taylor polynomials to integrate non-elementary functions via series, bound definite-integral errors, verify limits with Maclaurin series and L'Hôpital, and approximate cube roots with degree-two polynomials.

Explore vectors in space and their operations—summation, scalar multiplication, dot and cross products—and learn equations of lines, planes, and surfaces in three dimensional coordinates.

Explore three-dimensional coordinate systems by using the x, y, z axes and the right-hand rule, covering points, vectors, planes, octants, and the distance and magnitude formulas.

This lecture explores surfaces in R3, including cylinders and spheres. It uses completing the square to find centers and radii, and describes the region between spheres below the xy-plane.

Explore vectors in three-dimensional space by reviewing vectors in two dimensions, defining components, addition, subtraction, scalar multiplication, magnitudes, unit vectors, standard basis, and prep for dot and cross products.

Learn the dot product, the scalar product of two vectors in r3. Express it as a1b1 + a2b2 + a3b3, explore a·a = |a|^2, and a·b = |a||b|cos theta.

Explore projections of vectors onto another vector, including scalar and vector projections, using dot product, magnitudes, and cosine theta, with examples and related inequalities.

Explore the cross product of two vectors, its perpendicular result, determinant form, and key properties, including the right-hand rule, sign changes, and magnitude equals product of magnitudes times sine theta.

Learn to compute cross products using 3 by 3 determinants with i, j, k, apply the right-hand rule for direction, and use the cross product magnitude to find triangle area.

Describe a line in three-dimensional space using a point and a direction vector, then derive its vector, parametric, and symmetric equations. Apply to find the line's intersection with xy plane.

Describe a plane in space using a point and a normal vector, derive the scalar and linear equations via dot products, and compute the angle between planes from normal vectors.

Explore cylinders and quadric surfaces in space, defined by lines parallel to a given line and by three-variable equations. Examine traces and standard forms, including ellipsoids and elliptic paraboloids.

Explore vector functions describing a particle’s position in space, and learn to compute speed, velocity, and arc length, plus curvature and the Frenet-Serret frame.

Learn vector functions, where input t yields a vector r(t) with components f(t), g(t), h(t). See space and planar curves from parametric equations, including lines, unit circles, and helices.

Differentiate vector functions componentwise to obtain the tangent vector and unit tangent vector, then use secant vectors and tangent lines to connect derivatives with motion, with examples on a helix.

Compute the definite integral of a vector function by integrating its components and adding a vector constant when unbounded, illustrated with a helix r(t) from 0 to pi/2 yielding (2, 1, pi^2/4).

Compute the arc length of a space curve by integrating the speed, the magnitude of the derivative, from the bounds, illustrated with a circular helix.

Parametrize a space curve by arc length, compute s(t) from the magnitude of r'(t), solve t(s), then substitute to obtain r(s) from the starting point.

Define curvature as the rate of change of the unit tangent with respect to arc length, using a circle of radius a to show kappa equals 1/a.

Derives an alternate curvature formula for a space curve using the cross product of r' and r'' over |r'|^3, and demonstrates with a twisted cube example.

Derive the curvature of a planar curve y=f(x) using the second formula. Express kappa as |f''(x)|/(1+(f'(x))^2)^{3/2} and apply to y=x^2 to illustrate values.

Explore the Frenet-Serret frame as a moving coordinate system along a space curve, defined by the unit tangent t, the unit normal n, and the binomial vector b.

Explain the Frenet-Serret equations for the moving frame along a curve, with dt/ds = kappa n, dn/ds = -kappa t + tau b, and db/ds = -tau n.

Relate motion of a particle to a vector r(t); derive velocity v = r'(t), acceleration a = v'(t); compute speed |v| and obtain position by integrating velocity with initial conditions.

Decompose acceleration into tangential and normal components using unit tangent and unit normal vectors, curvature, and speed, with examples and Frenet-Serret concepts.

Explore multivariable functions that take multiple inputs and yield a single real value, covering limits, continuity, partial derivatives, tangent planes, higher derivatives, and linear approximation.

Explore multivariable functions that take multiple inputs and how input order matters. Learn about domains and independent versus dependent variables with area and volume examples and square-root constraints.

Visualize a function of two variables by plotting a surface z=f(x,y) in 3d space and by drawing level curves on the xy-plane, with examples f=6-3x-2y and g=sqrt(9-x^2-y^2).

Define two-variable limits with epsilon-delta, compare approaches along multiple paths, and illustrate nonexistence and existence cases using path examples and the squeeze theorem.

The lecture defines continuity for two-variable functions via the limit equaling the function value at (a,b), using a piecewise example to show continuity on R^2, including at the origin.

Explore partial derivatives of multivariable functions by fixing a variable, learn f_x and f_y notation, and apply chain rule and product rule through example calculations.

interpret fx and fy as the slopes of the intersection curves of the surface with planes y=b and x=a; for f(x,y)=4 - x^2 - 2y^2, fx(1,1)=-2 and fy(1,1)=-4.

Delve into higher derivatives of multivariable functions, deriving the four second partial derivatives and their notations, note the mixed derivative equality under continuity, with worked examples and tangent plane preview.

learn to compute the tangent plane to a two-variable surface at a point using partial derivatives, via z - z0 = f_x(x0,y0)(x - x0) + f_y(x0,y0)(y - y0).

Explore how tangent planes provide linear approximations for two-variable functions. Build L(x,y) from f(a,b) and its partials, verify differentiability, and apply to f(x,y)=x e^{xy} at (1,0) with L(x,y)=x+y.

Investigate differentials for single and multivariable functions, derive the total differential via partial derivatives, relate it to delta z through linearization on a tangent plane, and review sample with f(x,y)=x^2+3xy−y^2.

Explore the chain rule for single and multivariable functions, using partial derivatives and a tree diagram to compute dz/dt via x(t) and y(t).

Explore the chain rule for multivariable functions using tree diagrams to compute total derivatives with respect to t, through three examples.

Explore implicit differentiation for multivariable functions, deriving dy/dx and dz/dx from f_x, f_y, and f_z, with x^3+y^3=6xy and x^3+y^3+z^3+6xyz as examples.

See how the directional derivative equals the gradient magnitude when the unit vector aligns with the gradient, revealing the maximum rate of change.

Use the gradient of f to define the normal to the tangent plane of the level surface f(x,y,z)=k, then form the tangent plane and normal line, illustrated on an ellipsoid.

Discover how to locate max and min values for two-variable functions by finding critical points, applying the second derivative test, and identifying saddle points using the two by two determinant.

Explore maximum and minimum values by solving two-variable optimization problems: minimize distance to a plane via distance squared and maximize volume of a lidless box from a fixed surface area.

Analyze absolute maximum and minimum values of a function of two variables on a closed region by the extreme value theorem, locating interior critical points and evaluating boundary segments.

Apply the method of Lagrange multipliers to maximize or minimize a function under a constraint. Solve for x and y and z using a lambda multiplier when gradients are parallel.

Apply Lagrange multipliers to maximize volumes and quadratic forms under constraints, solving open-box with 12 m² surface area and a circle constraint x^2+y^2=1.

Apply Lagrange multipliers with two constraints to maximize or minimize a multivariable function by equating its gradient to a combination of constraint gradients, illustrated on a plane and a cylinder.

Explore multiple integrals for functions of several variables and the volumes under their surfaces. Learn double Riemann sums and double integrals over a rectangular region.

Learn to estimate and compute the volume under a surface using double Riemann sums and double integrals over rectangles, by subdividing into small rectangles and taking a limit.

Learn to estimate the volume under a surface with double integrals over a region using rectangle methods. Compare upper-right corner and midpoint sampling, and see how finer grids improve accuracy.

Learn how to compute the average value of a function and apply it to a region to relate area, volume, and double integrals for estimating the volume under a surface.

Learn to evaluate iterated integrals by performing inner and outer integrations, treating the other variable as constant, and observe how the order of integration can yield the same result.

Apply Fubini's theorem to show the double integral over a rectangle can be evaluated in any order. Two examples show same result for f(x,y)=x−3y^2 on 0≤x≤2,1≤y≤2 and 16−x^2−2y^2 on 0≤x≤2,0≤y≤3.

Explore how f(x,y)=g(x)h(y) makes a double integral over a rectangle a product of two single-variable integrals. A sine-cosine example over [0, pi/2]×[0, pi/2] yields 1.

Explore evaluating double integrals over general regions using type one and type two bounds, with vertical or horizontal subdivisions, to compute the volume under the surface.

Explore five examples of evaluating double integrals over general regions using type one and type two methods. Learn to switch integration order and use vertical and horizontal rectangles.

Apply properties of double integrals to split sums, pull constants, and compare surfaces. Learn to partition regions and bound integrals using area and constant bounds, with a disk example.

Learn to evaluate double integrals with polar coordinates, describe regions using r and theta, recognize polar rectangles, apply the Jacobian r, and transform from rectangular to polar coordinates.

Learn to evaluate double integrals in polar coordinates over general regions. Analyze area of one loop of r=2 cos 2 theta and volume under z=x^2+y^2 inside x^2+y^2=2x.

Break the surface into tiny rectangles and derive the surface area using the cross product; evaluate the double integral over region D of sqrt(1 + f_x^2 + f_y^2).

Celebrate finishing the course by leaving a review to help future students and share feedback to improve the course, and wish you good luck for your future study.