NYB) Mathematics Department, Vanier College

17/01/2016 Calculus II for Science (NYB) Mathematics Department, Vanier College

Content Performance Criteria* Competency No. 1: To determine the indefinite integral of a function. Topic: ANTIDERIVATIVES 1.1 Define antiderivative, indefinite integral, constant of integration and give the properties of indefinite integrals. 1 , 7 , 8 Give basic indefinite integrals involving algebraic, trig, log, exponential and inverse trig functions (i.e. those following directly from the 1.2 1 , 3 , 8 derivatives). 1.3 Find antiderivatives satisfying certain boundary conditions. 1 , 3 , 5 , 6 , 7 , 8 Topic: DIFFERENTIALS 1.4 Give the definition of the differential. 1 , 8 1.5 Give the geometric interpretation of the differential and use it to make linear approximations. 1 , 3, 5 , 6 , 7 , 8 Topic: TECHNIQUES OF INTEGRATION 1.6 Perform algebraic substitutions. 1 , 3 , 4 , 5 , 8 1.7 Integrate trig functions, their powers and combinations, using trig identities where necessary. 1 , 3 , 4 , 5 , 8 1.8 Find integrals involving or yielding exponential and log functions 1 , 3 , 4 , 5 , 8 1.9 Find integrals using integration by parts, partial fractions, completing the square, trig substitution or a combination of these. 1 , 3 , 4 , 5 , 8 1.10 Use the above techniques to find integrals involving or yielding inverse trig., exponential and logarithmic functions. 1 , 3 , 4 , 5 , 8

Competency No. 2: To calculate the definite integral and the improper integral of a function in an interval. Topic: THE DEFINITE INTEGRAL 2.1 Use the sigma notation to write a sum in closed form. 1 , 8 2.2 Expand a sum written in the sigma notation. 1 , 3 , 8 2.3 State and apply the properties of the sigma notation. 1 , 8 푛 푛 2 푛 3 2.4 State and apply one or more of the formulas for, ∑𝑖=1 푖, , ∑𝑖=1 푖 , and , ∑𝑖=1 푖 . 1 , 3 , 5 , 8 2.5 Calculate a Riemann sum using summation formulas. 1 , 3 , 5 , 7, 8 2.6 Give the definition of the definite integral and evaluate a definite integral using a Riemann sum. 1 , 3 , 5 , 7, 8 2.7 Demonstrate an understanding of the definite integral as an accumulation of infinitesimal quantities. 1 , 8 2.8 Give the properties of the definite integral. 1 , 8 2.9 Interpret the definite integral for positive and negative functions. 1 , 2 , 8 2.10 Demonstrate an understanding of both versions of the Fundamental Theorem of Calculus. 1 , 2 , 8 Calculate a definite Integral using the Fundamental Theorem of Calculus, including an integral involving a change of variable (and new limits of 1 , 3 , 5 , 8 2.11 integration) 2.12 Demonstrate an understanding of the Mean Value Theorem for definite integrals. 1 , 8 2.13 Use the techniques of integration listed above to evaluate definite integrals. 1 , 3 , 4 , 5 , 8 2.14 Perform numerical integration using the Trapezoid Rule, Simpson’s Rule or an appropriate Taylor series. 1 , 3 , 4 , 5 , 8 Topic: IMPROPER INTEGRALS Determine the divergence or convergence of an integral in the following cases: a) At least one of the limits of integration is not a real number. 2.15 1 , 3 , 4 , 5 , 7, 8 b) There is a discontinuity within the limits of integration. c) Both a) and b).

Competency No. 3: To calculate volumes, areas and lengths and draw two- and three-dimensional representations. To express concrete problems as differential equations and solve simple

differential equations. Topic: APPLICATIONS OF THE DEFINITE INTEGRAL 3.1 Find the area of a region in the plane using horizontal of vertical slices and judge which method is most efficient for a particular problem. 1 , 2 , 3 , 4 , 5 , 6 , 8 3.2 Find the average value of a function on an interval. 1 , 3 , 4 , 5 , 7 , 8 3.3 Find volumes of revolution using the method of disks or cylindrical shells and judge which method is most efficient for a particular problem. 1 , 2 , 3 , 4 , 5 , 6 , 8 3.4 Calculate arc length. 1 , 2 , 3 , 4 , 5 , 7 , 8 3.5 Calculate surface area. 1 , 2 , 3 , 4 , 5 , 7 , 8 3.6 Calculate work and pressure. 1 , 2 , 3 , 4 , 5 , 7 , 8 3.7 Mathematically formulate a situation involving a differential equation. 1 , 2 , 3 , 4 , 5 , 7 , 8 3.8 Solve separable differential equations including exponential growth and decay problems. 1 , 2 , 3 , 4 , 5 , 7 , 8

17/01/2016 Calculus II for Science (NYB) Mathematics Department, Vanier College

Competency No. 4: To analyze convergence of series. Topic: INFINITE SERIES 4.1 Recognize arithmetic and geometric sequences and find a formula for the 푛푡ℎ term. 1 , 9 4.2 Find a formula for the nth term of a given sequence defined by a rational function. 1 , 5 , 6 , 9 4.3 Determine the divergence or convergence of a given sequence of the above types. 1 , 5 , 6 , 9 4.4 Determine the convergence or divergence of a sequence. Use the Squeeze Theorem to calculate the limit of a sequence. 1 , 3 , 8 4.5 Define special series: geometric series, p-series, harmonic series, telescoping series. 1 , 8 4.6 Define the convergence or divergence of infinite series. 1 , 8 4.7 Determine the convergence or divergence of geometric series and telescoping series and find the sum if convergent. 1 , 3 , 7 , 8 Determine convergence or divergence of series using: Test for Divergence, Integral Test, Comparison Test, Limit Comparison Test, Ratio Test, 1 , 3 , 7 , 8 4.8 Root Test 4.9 Define absolute and conditional convergence for series. 1 , 8 4.10 Apply alternating series test. 1 , 3 , 5 , 7 , 8 4.11 Define and utilize radius and interval of convergence. 1 , 8 4.12 Use operations on power series. Find power series by differentiation and integration. 1 , 5 , 7 , 8 1 , 3 , 5 , 7 , 8 4.13 Find Taylor and MacLaurin series for 푠푖푛푥, cos(푥) , arctan(푥) , 푒푥, ln(1 + 푥) , (1 + 푥)푛 and related functions. 4.14 Estimate remainder, determine the radius of convergence. 1 , 3 , 5 , 7 , 8

* Performance Criteria

1: Appropriate use of concepts. 2: Representation of a situation as a function. 3: Accurate graphical representation of a function. 4: Correct choice and application of differentiation techniques. 5: Use of algebraic operations in conformity with rules. 6: Accuracy of calculations. 7: Correct interpretation of results. 8: Explanation of steps in problem-resolution procedure. 9: Use of appropriate terminology

Remarks:

1) The numbering of the content in this document is merely for reference purposes. The actual order in which the course material is presented is at the teacher's discretion. 2) At the teacher's discretion, appropriate and closely related definitions, derivations, proofs and applications using pertinent technology may be added and form part of the evaluation.