Learning objectives for MATH 1552

Transcription

1 Learning objectives for MATH 1552 Course learning objectives 1. Students will master basic Calculus concepts, including integration techniques, convergence of integrals and infinite series, and Taylor s theorem. 2. Knowledge of the above concepts will be exhibited algebraically and geometrically. 3. Calculus concepts will be applied to solve physics, geometry, and numerical approximation problems. 4. Students will understand the usage of proper mathematical notation in relation to the above topics. Unit topics with quiz and midterm schedule (I) For Quiz ,5.2 Riemann sums and sigma notation The definite integral. (II) For Midterm 1 1. Ch. 5, 7.1,7.2,8.2. Integration by substitution, area between curves, separable DEQs, integration by parts, the Fundamental Theorem of Calculus. (III) For Quiz ,8.4 Trig integrals and trig substitution Partial fractions decomposition. (IV) For Midterm 2 1. Ch. 8, 4.5,10.1,10.2. Numerical integration, L Hopital s Rule, improper integrals, sequences and infinite series. 1

2 (V) For Quiz ,10.4 Integral test, comparison tests Ratio and root test. (VI) For Midterm 3 1. Ch. 10. Alternating series, power series, Taylor and MacLaurin Series. (VII) For Quiz Advanced Taylor series ,6.2 Volumes by disks and shells. Daily learning objectives Week 1: Review of Derivatives/Anti-derivatives, Area Under the Curve, Sigma Notation 4.8: Antiderivatives 1. Memorize the definition of antiderivative on page 281. Definition: A function F is an antiderivative of f on an interval I if F (x) = f(x) for all x in I. 2. Memorize and practice utilizing the formulas in Table 4.2 pg. 282 in 4.8, the antiderivative formulas. 2

3 3. Understand the reason for +C in the general antiderivative, accompanied by pictures of several functions who differ by a constant f(x) + C and noticing that the slopes of all the functions are the same. 5.1 & 5.2: Area Under the Curve Sigma Notation 1. Be able to compute left-endpoint, right-endpoint, upper-estimate, and lowerestimate finite sum approximations to areas under a curve y = f(x) using n rectangles. 2. Be able to compute a midpoint approximation of the area under a curve y = f(x) using n rectangles. 3. Practice using and recognizing sigma notation to express finite sums. Week 2: The Definite Integral, The Fundamental Theorem of Calculus 5.3: The Definite Integral 1. Understand how to use a sigma notation for a Riemann Sum in order to reconstruct the definite integral that is being approximated (similar to problems 1-8 on page 324 in 5.3). 2. Memorize the formulas for left-endpoint and right-endpoint approximations of definite integrals using Riemann Sums in Sigma notation. 3. Know and understand the sum, difference, constant multiple, and constant value rule for finite sums in Sigma notation. 4. Find a closed form for a finite sum using the Gauss formula n i=1 k = n(n+1) 2 and the formulas on pg. 312 for the first n squares and the first n cubes. Quiz #1: 5.1,5.2 Riemann sums and sigma notation and 5.3 The definite integral. 5.4: The Fundamental Theorem of Calculus 1. Memorize the Fundamental Theorem of Calculus Part I & II, Theorem 4 on page

4 2. Be able to compute definite integrals with the FTC, using correct notation and clearly showing all steps. 3. Be able to compute F (x) given F (x) expressed as a definite integral in the variable t with one of the limits as x or a function f(x). The latter uses the chain rule, and the student should attempt to understand the proof of the general form of the FTC part I. Week 3: Curves Integration by Substitution, Area Between 5.5: Integration by Substitution 1. Set up the substitution box or other learning tool which has two parts u and du. For example in the box could be u = 4x 2 and du = 8x dx. 2. Gain experience choosing the u, and being able to find the du by differentiation, including the dx term for accurate substitutions. 3. Be able to see what can go wrong if the wrong u is chosen, how it is not possible to pull out functions of x from an integral if there are extra terms. 5.6: Area Between Curves 1. Be able to find intersection points of two or more curves by setting y = y and then solving for x. 2. Set up and use a sign chart to determine which function is top and which is bot. 3. Understand the meaning of the formula A = b a top bot dx, and how it is used to calculate area between the curves bot top over the interval [a, b]. 4

5 Week 4: Logs, Exponentials and Separable DEQs, Integration by Parts 7.1,7.2: Logs, Exponentials and Separable DEQs 1. Be able to separate a separable DEQ, integrate both sides, and solve for y. 2. Be able to solve an initial value problem by finding C given an initial condition. 3. Understand the problem of reusing C in an initial value problem, if you plug into one formulation to find C and then use that C in a different equation where the C has changed. 8.2: Integration by Parts 1. Memorize the integration by parts formula u dv = uv v du 2. Memorize the ILATE acronym (Inverse trig, logs, algebra, trig, and exponential functions) and be able to recognize the parts of the integrand as each type. 3. Recognize an integral as a potential integration by parts integral by being able to perform mental math to quickly notice the du and v parts of the integration by parts technique. 4. Gain experience and intuition to realize when the u and dv have been chosen incorrectly because the integral is getting worse and not better after applying the IBP formula. 5. Be able to perform a wrap around integration by parts problem such as sin(x)e x dx where the integral appears after two iterations of integration by parts and then can be solved for. 6. Be able to compute a definite integral using the correct notation to evaluate uv in the IBP formula. 5