M408D Class Outlines for l Hospital Rule Integrals and Sequences Review and Problem Solving Skills Qimou Su qimousu math utexas edu Math University of Texas at Austin June 24 2004 1 Evaluate Function s Limits Using l Hospital Rule A Recipe for Cooking Limits using l Hospital Rule Step 1 Recognize the 7 indeterminate forms 0 0 Step 2 Convert to the basic indeterminate types 0 0 0 0 00 and 1 or Step 3 Apply the l Hospital Rule to evaluate the limit Taking derivatives on both numerator and denominator Step 4 Check if the limit is determinate If so the food is ready If the limit is still indeterminate goes to Step 1 For more details and examples refer to my class notes of 06 15 04 2 Integration Techniques and Improper Integral 2 1 Integration Techniques Algebraic Skills try to decompose or expand the integrand into small pieces and integrate them separately Substitution simple substitution formal substitution Trigonometric substitution Integration by Parts use the order I L A T E InverseRTrigonometric Log Algebraic TrigonoR metric Exp to make the choice for u udv uv vdu Partial Fraction Decomposition decompose the rational function into partial fractions and integrate them separately Refer to my class notes on 06 17 04 for details 1 2 3 M408D Class Outlines for l Hospital Rule Integrals and Sequences Evaluate the Improper Integral Problem Type 1 Check if the integrand has irregular points Is the integrand unbounded 2 Find the anti derivative indefinite integral using the integration techniques listed above 3 Evaluate the definite integral using Fundamental Theorem of Calculus 4 Take limit to evaluate the improper integral Refer to section 8 8 of textbook and my class notes for details 4 Find a Formula for General Term of a Sequence Problem Type 1 Pattern recognition like problem 9 14 on the textbook P 747 Refer to my class note on 06 22 04 for details Problem Type 2 Find a formula for general term for sequence represented by a recursive equation e g suppose sequence an satisfies that a1 3 and an 1 2an 1 Try to find a formula for general term an Refer to my class note on 06 22 04 for the solving skills 5 Determine the Monotonicity of Sequences Here are some effective skills for solving this kind of problems Using the monotonicity of functions good for simple an For complex bn check the sign of the derivative e g an ln n or bn ne n Compare the difference an 1 an to 0 good for Algebraic an e g an n 1 n Compare the ratio an 1 an to 1 if an 0 for all n good for factorial and exponential an e g an nn n Refer to my class notes on 06 22 04 for details 6 Boundedness of Sequences Refer to my class notes on 06 22 04 for details
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