Complete Math Course: From Functions to Calculus & AI

Explore how right triangles model distances and angles using sine, cosine, and tangent, derive angle measures from ratios, and apply area formulas for triangles and squares.

Introduces the second function, the co-second function, and the co-tengent function, defining them as 1/cos x, 1/sin x, and sin x over cos x, showing their relation to tangent.

Derive and apply the sine and cosine addition formulas and the tangent addition formula, exploring negative angle cases and the interplay of sine, cosine, and tangent.

Explore trig identities by deriving sin(alpha+pi) equals minus sin(alpha) and cos(alpha+pi) equals minus cos(alpha). Demonstrate sin(pi-alpha) equals sin(alpha) and explain the 2pi period and 180-degree relation using graphs and expressions.

Explore limits, asymptotes, and quotient behavior as x tends to infinity or zero, including x+1, 1/x, and f(x)/g(x) limits with equal growth rates.

examine limits through three cases: a linear function yields 11 at x=3, a zero-over-zero rational expression cancels to 4, and a zero-over-zero square-root case yields 1/4.

Explore the natural logarithm and its relation to exponentials, ln, and base-a logs, with applications in sound decibels, pH, and AI information theory.

Explore real-life uses of logarithms, from decibels and the Richter scale to entropy, log loss, and softmax in AI and neural networks.

Define derivatives as the limit of F(X+H) minus F(X) over H to measure change. Show that the derivative of 3X+4 is 3 and that of X squared is 2X.

The lecture applies the power rule to polynomials by differentiating each term of f(x). It shows constants yield zero and the example f(x)=5x^4+7x^3+4x^2+8x+5 has derivative f'(x)=20x^3+21x^2+8x+8.

Derive the derivative of sin x using the limit definition, showing sin x differentiates to cos x, and establish that the derivative of cos x is -sin x.

Learn how to differentiate exponential functions using the chain rule, including e^{g(x)} with g(x)=ax^2+bx+c, and differentiate logarithmic forms such as ln x and (ln x)^2.

Apply the quotient rule to h(x) = f(x)/g(x) and compute h'(x) = (f'g - fg')/g^2. Use examples with f(x)=sin x and g(x)=cos x to get h'(x)=sec^2 x, the derivative of tan x.

Explore how three-dimensional functions defined by f(x,y) assign elevation from latitude and longitude to shape landscapes, roller coasters, and economic visuals of supply, demand, and price.

Compute partial derivatives of z = x^2 + y: ∂z/∂x = 2x, ∂z/∂y = 1. Define the gradient as sqrt((∂z/∂x)^2+(∂z/∂y)^2) and the Laplacian as described, yielding sqrt(1+4x^2).

discover the optimization algorithm behind ai, showing how updating r by subtracting the learning rate times the gradient lowers the loss and cost over epochs.

Explore differentiation and integration basics, including constants, linear functions, and primitives, then compute the area under a line y = 2/7 x + 2 using the integral and area decomposition.

Derive that ∫ cos x dx = sin x + C and ∫ sin x dx = -cos x + C, verified by the derivatives of sine and cosine.

Learn integration by parts, derive ∫ u dv = uv - ∫ v du from the product rule, and apply it to x e^x to compute the integral.

Learn to integrate h(x)=f(x)/g(x) using u-substitution, yielding ln|g(x)|+C; see examples with 1/x and g(x)=x, and with h(x)=2x/(x^2+1) or x/(x^2+5).

Master integration by parts to derive ∫ ln x dx and ∫ (ln x)/x dx: x ln x − x + c and (ln x)^2/2 + c.

Compute the integral of arctan(x^2+1) using integration by parts, setting u = arctan(x^2+1) and dv = dx, yielding x arctan(x^2+1) - 2x + 2 arctan x + C.

Explore hyperbolic functions, including sinh x, cosh x, and tanh x, defined with exponentials, with inverses and the key identity cosh^2 x minus sinh^2 x equals 1.

Explore integrals of hyperbolic functions, showing that the integral of the apropolic sine of x equals the apropolic cosine of x plus c.