Understanding Calculus - Course 2

Explain the idea of differentiation via secant lines through (x, f(x)) and (x+h, f(x+h)), define the derivative as the limit of slope, and derive the tangent line.

Show that differentiability at a point implies continuity at that point, making differentiable functions automatically continuous and illustrating an epsilon-delta proof approach to test continuity.

Explore how derivatives apply to algebraic operations, including sum, difference, constant multiple, product, and quotient rules for differentiable functions f and g at a point.

Explore the chain rule for differentiating composite functions, defining the composition and proving that (g∘f)'(x0) = g'(f(x0)) f'(x0) under suitable conditions.

explain when a function is invertible (one-to-one and onto) and show that if f is differentiable with nonzero derivative, its inverse is differentiable and (f^{-1})'(f(x0)) = 1/f'(x0).

this lecture proves that polynomial functions are differentiable everywhere and presents the power rule: derivative of x^n is n x^{n-1}, with constants differentiating to zero, and linearity of differentiation.

Derives sine, cosine, and tangent derivatives using geometry, and establishes key limits sin x goes to 0, cos x goes to 1, and sin x over x goes to 1.

This lecture presents derivatives of trigonometric functions, proves d/dx sin x = cos x and d/dx cos x = -sin x, and derives tan x and sec^2 x.

Explore derivatives of exponential functions with base a and the natural e^x. Relate logarithms as inverses and derive d/dx a^x equals a^x ln a and d/dx ln x equals 1/x.

Explore derivatives of power functions f(x)=x^alpha with domain x>0, using the chain rule and logarithmic identities. Differentiate to obtain d/dx x^alpha = alpha x^{alpha-1}.

Explore how to differentiate complex functions with chain rule, product rule, and quotient rule, illustrated through many examples of cosine, sine, exponential, and logarithmic expressions.

Apply the first derivative test to locate local maxima and minima, define local extrema, and confirm that differentiable points with a local extremum satisfy f'(c)=0.

Explore the mean value theorem: a function continuous on [a,b] and differentiable on (a,b) has a c where f'(c) equals the secant slope; illustrated via graphs and Rolle’s theorem.

Explore how derivatives indicate monotonicity: f' sign dictates increasing or decreasing behavior, with strict versus non-strict cases and examples like x^3.

apply differentiation to sketch functions by using the first derivative test to locate where they increase or decrease. illustrate with a degree-four polynomial, finding critical points and sketching the graph.

Learn how the second derivative test identifies local extrema: if f''(c) > 0, a local minimum; if f''(c) < 0, a local maximum; if f''(c) = 0, inconclusive.

Apply l’Hôpital’s rule to evaluate indeterminate limits such as 0/0 and infinity/infinity using derivatives. Repeat the process under suitable conditions to resolve limits, with examples like e from (1+x)^(1/x).

Explore how to apply l'Hôpital's rule to evaluate limits, including 0/0 and infinity forms, with examples like sin x / x and log x over e^(1/x).