Option:Calculus

• Informal treatment of limit of sum, difference, product, quotient; squeeze theorem.
• Divergent is taken to mean not convergent.

• Convergence of infinite series.
• Tests for convergence: comparison test; limit comparison test; ratio test; integral test.
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• Series that converge absolutely.
• Series that converge conditionally.
• Alternating series.
• Power series: radius of convergence and interval of convergence. Determination of the radius of convergence by the ratio test.

The sum of a series is the limit of the sequence of its partial sums.

Students should be aware that if  then n the series is not necessarily convergent, but if  the series diverges.

is convergent for p >1 and divergent otherwise. When p = 1, this is the harmonic series. Conditions for convergence.

The absolute value of the truncation error is less than the next term in the series.

• Continuity and differentiability of a function at a point.
• Continuous functions and differentiable functions.

• The integral as a limit of a sum; lower and upper Riemann sums.
• Fundamental theorem of calculus.
• Improper integrals of the type.

• Int: How close was Archimedes to integral calculus?
• Int: Contribution of Arab, Chinese and Indian mathematicians to the development of calculus.
• Aim 8: Leibniz versus Newton versus the “giants” on whose shoulders they stood—who deserves credit for mathematical progress?
• TOK: Consider 
• An infinite area sweeps out a finite volume. Can this be reconciled with our intuition? What does this tell us about mathematical knowledge

?

• First-order differential equations.
• Geometric interpretation using slope fields, including identification of isoclines.
• Numerical solution of using Euler’s method.
• Variables separable.
• Homogeneous differential equation
• using the substitution y = vx.
• Solution of y′ + P(x)y = Q(x), using the integrating factor.

• Rolle’s theorem. Mean value theorem.
• Taylor polynomials; the Lagrange form of the error term.
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• Use of substitution, products, integration and differentiation to obtain other series.
• Taylor series developed from differential equations.

• Applications to the approximation of functions; formula for the error term, in terms of the value of the (n + 1)^th derivative at an intermediate point.
• Students should be aware of the intervals of convergence.

• Int, TOK: Influence of Bourbaki on understanding and teaching of mathematics.
• Int: Compare with work of the Kerala school.

The evaluation of limits of the form

Using l’Hôpital’s rule or the Taylor series.