Download Test #2 Review

MAC 2311 – CALCULUS I
REVIEW FOR TEST #2
FLORIDA INTERNATIONAL UNIV.
SPRING 2012
REMEMBER TO BRING AN 8x11 BLUE EXAM BOOKLET FOR THE TEST
Relevant sections of the textbook: [Ch.2: Sec 2.3-2.6,
Ch.3: Sec. 3.1-3.6,
Ch.4: Sec.4.1-4.2]
MAIN PROBLEM SOLVING TECHNIQUES:
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How to find the derivatives of polynomials and rational functions by using the Rules of
Differentiation (Sum Rule, Product Rule, Reciprocal Rule, and Quotient Rule).
How to find derivatives of sin(x) & cos(x) directly from the definition and of rational
expressions involving trigonometric functions by using the Rules of Differentiation.
How to find derivatives by using the Chain Rule in all three forms.
Implicit differentiation, i.e., how to find the derivatives of implicitly defined functions.
How to find the derivative of ln(x) and how to use this in logarithmic differentiation.
How to find derivatives of inverse functions, of ex, and of inverse trigonometric functions.
How to solve word problems involving rates of change by using the Chain Rule, and so on.
How to use differentials to estimate an expression and to solve problems with relative errors.
How to find limits by using L’Hospital’s 0/0 & ∞/∞ Rules and by the “ln trick”
How to find the turning points and intervals of increase & decrease by using the first
derivative. How to find the inflection points and the intervals of upward & downward
concavity by using the second derivative.
How to find the local (relative) maxima & minima of functions and how to sketch the graph
of a function using the information found in #10 above.
KEY CONCEPTS AND MAIN DEFINITIONS:
Rate of change of one quantity with respect to another, Differential of a function, Local
linear approximation, Relative error, Measurement error ∆x, Propagated error ∆y, 0/0 & ∞/∞
forms, Other indeterminate forms, The “ln trick”, Intervals of increase & decrease, Intervals of
upward & downward concavity, Stationary points, Turning points, No-curvature points, Inflection
points, Local (relative) Maximum & Minimum, First-derivative Test, Second-derivative Test.
MAIN FORMULAS AND THEOREMS:
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[f -1(x)]’ = 1 / [f ’(f -1(x)]
(f+g)’(x) = f ’(x) + g’(x), (f-g)’(x) = f ’(x) - g’(x), (f.g)’(x) = f ’(x).g(x) + f(x).g’(x).
(c.f)’(x) = c.f ’(x), (1/g)’(x) = - g’(x)/[g(x]2, (f/g)’(x) = {f ’(x).g(x) - f(x).g’(x)} / [g(x]2
(fog)’(x) = f ’(g(x)).g’(x), dy/dx = (dy/du).(du/dx), (d/dx){f(u)} = [(d/du){f(u)}].(du/dx)
sin’(x) = cos(x), tan’(x) = [sec(x)]2, sec’(x) = sec(x).tan(x), ln’(x) = 1/x, exp’(x) = exp(x)
[sin-1(x)]’ = 1 / √{1- x2}, [tan-1(x)]’= 1 / {1+ x2}, [sec-1(x)]’ = 1 / [x.√{x2-1}].
If y = f(x), then ∆y ≈ f ’(x0).∆x and f(x0+∆x ) ≈ f(x0) + f ’(x0).∆x locally near x0
limx→a {f(x)/g(x)} = limx→a {f ’(x)/g’(x)} if limx→a f(x)/(g(x)) is of 0/0 or ∞/∞ form.
Indeterminate forms: 0/0, ∞/∞, 0.∞, ∞.0, (1)∞, (0)0, (∞)0, ∞ - ∞.
Stationary points are points where f ’(x)=0, No-curvature points are points where f ’’(x)=0
Turning points are points where f changes from incr. to decr., or from decr. to increasing.
Inflection points are points where f changes its concavity from up to down or down to up.