Math 110 Fall 2024 Study Guide for Exam II PDF

MATH 110 Fall 2024 Don Lawrence Study Guide for Exam II The exam will cover homeworks 6 through 9. No notes, books, internets, devices, calculators, other people, or other people’s exams. I try to make study guides complete, but there is no guarantee – I might think of something lat...

MATH 110 Fall 2024 Don Lawrence Study Guide for Exam II The exam will cover homeworks 6 through 9. No notes, books, internets, devices, calculators, other people, or other people’s exams. I try to make study guides complete, but there is no guarantee – I might think of something later that I forgot to include. Also look at homeworks, quizzes, handouts, labs, and class notes. Topics Derivatives o Definition, and “differentiable” o The three common examples of nondifferentiability in graphs o Notations o Compute using the definition o Use the definition of derivative to prove basic differentiation formulas such as ( f ( x) + g ( x)) = f ( x) + g ( x) o Know all differentiation rules and formulas (through (arctan x)  on the handout) o The units on a derivative are always function units variable units o To predict/explain the sign of a derivative, increase the variable; does the function increase or decrease? o To compute f (a ) (except from the definition), first compute f (x) , then plug in x = a o Know the proof of the product rule, including the explanations at the end o Prove deriv formulas for the trigs besides sin and cos, using the quotient rule and trig identities o Implicit Differentiation o Logarithmic Differentiation, including using it to prove the deriv formulas for e x , a x , and x n o The relationships among position, velocity, and acceleration o “Symbolic Differentiation”; e.g., Section 2.5, # 61, 76 Inverse functions o They cancel each other (e.g., when solving equations) o Only 1-1 functions have inverses o Given the graph of f , sketch the graph of f −1 o Simplify any trig(arctrig(x)) o For a given trig function, restrict the domain to a maximal 1-1 arc (or arcs!), sketch the graph of the inverse of that arc, and find and simplify the derivative of that inverse trig function o For “pseudo-inverse functions”, such as all of the inverse trig functions, only one of the cancelations formulas works for all inputs Concavity o Given by f  o Determines whether a tangent line is likely to produce an overestimate or underestimate Related rates Optimization o Find critical points. There won’t be any critical points of the f  DNE type, but you should know that those are included in the definition of “critical point” o Classify them using the first derivative test o Classify them using the second derivative test o For a (one-variable) continuous function, if there’s only one critical point that’s a max or min, then it’s the global max or min.