CALCULUS 1

  1. ##### Lesson 12: Techniques of Antidifferentiation 1.1. Topic 12.1: Antidifferentiation by Substitution and by Table of Integrals

    • Substitution method (u-substitution):

      • If you have an integral of the form \int f(g(x)) g'(x) dx , set u = g(x) so that du = g'(x) dx and the integral becomes \int f(u) du.

      • For definite integrals, change the limits: \inta^b f(x) dx = \int{u(a)}^{u(b)} f(x(u)) x'(u) du.

    • Table of Integrals: use standard antiderivatives to shortcut evaluation, e.g.

      • \int x^{n}\,dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1

      • \int \frac{dx}{x} = \ln|x| + C

      • \int e^{ax}\,dx = \frac{1}{a} e^{ax} + C

      • \int \sin(bx)\,dx = -\frac{1}{b}\cos(bx) + C

      • \int \cos(bx)\,dx = \frac{1}{b}\sin(bx) + C

    • Significance: substitution reduces complex integrals to simpler forms; tables provide quick access to common antiderivatives.

    • Practical notes: recognize patterns that fit substitution; verify by differentiating the result.

    • Connections: links to the Fundamental Theorem of Calculus (FTC) since antiderivatives are the reverse of derivatives; underpins area calculations via definite integrals.

  2. ##### Lesson 13: Application of Antidifferentiation to Differential Equations 2.1. Topic 13.1: Separable Differential Equations

    • Form: dy/dt = g(t) h(y) (variables separated: put y terms on one side and t terms on the other).

    • Separation: dy/h(y) = g(t) dt

    • Integration: \int dy/h(y) = \int g(t) dt + C

    • Solve for y(t) using initial condition y(t0) = y0 if given; may require algebraic manipulation to isolate y.

    • Example template: if dy/dt = k y, then \int dy/y = \int k dt \rightarrow \ln|y| = kt + C \rightarrow y = C e^{kt}.

    • Practical use: models where a rate depends on both the current state and an independent variable (e.g., time).

    • Significance: shows how antidifferentiation techniques solve simple first-order ODEs.

  3. ##### Lesson 14: Application of Differential Equations in Life Sciences 3.1. Topic 14.1: Situational Problems Involving Growth and Decay Problems

    • Common model: exponential growth/decay dP/dt = r P, yielding P(t) = P_0 e^{r t}.

    • Acknowledge carrying capacity variants (e.g., logistic model) when appropriate: dP/dt = r P (1 - P/K).

    • Procedure: translate real-world situation into a differential equation, solve via separation or known forms, apply initial data, interpret results.

    • Relevance: helps in population biology, pharmacokinetics, cell growth, radioactive decay, and epidemiology.

  4. ##### Lesson 15: Riemann Sums and the Definite Integral 4.1. Topic 15.1: Approximation of Area using Riemann Sums

    • Partition [a,b] into n subintervals, choose sample points xi^* in each subinterval, \Delta xi = (b-a)/n (for equal partitions).

    • Riemann sum: Sn = \sum{i=1}^{n} f(xi^*) \Delta xi.. The definite integral is the limit: \inta^b f(x)\,dx = \lim{n\to\infty} S_n.

    • Important ideas: choice of sample points leads to left/right/midpoint sums; as n\to\infty, all converge to the same limit when f is integrable.
      4.2. Topic 15.2: The Formal Definition of the Definite Integral

    • Definition: \inta^b f(x)\,dx = \lim{\parallel P \parallel\to 0} \sum{i=1}^{n} f(xi^*) \Delta xi. where P is a partition of [a,b] with mesh \parallel P \parallel = \max \Delta xi.

    • Domains and integrability: f must be integrable on [a,b]; if f \geq 0, the integral represents area under the curve.

    • Significance: formal bridge between geometry (area) and analysis (limits); lays groundwork for FTC.

  5. ##### Lesson 16: The Fundamental Theorem of Calculus 5.1. Topic 16.1: Illustration of the Fundamental Theorem of Calculus

    • Part 1 (FTC1): If F is an antiderivative of f on [a,b], then \int_a^b f(x)\,dx = F(b) - F(a).

    • Consequence: differentiation and integration are inverse processes