MAC 2311 – Calculus I
REVIEW FOR TEST #1
FLORIDA INTERNATIONAL UNIV.
SPRING 2012
REMEMBER TO BRING AN 8x11 BLUE EXAM BOOKLET FOR THE TEST
Relevant sections of the textbook: [Ch.0: Sec 0.1-0.5,
Ch.1: Sec. 1.1-1.6,
Ch.2: Sec.2.1-2.4]
MAIN PROBLEM SOLVING TECHNIQUES:
1.
How to find the natural domains of functions, how to sketch the graph of a function f,
and how to use it to find the range of f
2.
How to find the compositions fog and gof and how to find the domains of the fog and gof.
3.
How to find the inverse, f -1(x) and dom(f -1), of a given one-to-one function f(x).
4.
How to simplify expressions, and how to solve equations, involving ex, ln(x), and x.
5.
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How to find limits as x→ a, x→ a+, x→a-, x→ + ∞, and x→ - ∞ by using Common
sense, the Theorem on Limits, Continuity, and other Special Techniques.
How to prove that limit of f(x) as x→ a is L, by using the epsilon-delta method.
How to find limits involving trigonometric, logarithmic and exponential functions.
How to use the Intermediate Value Theorem to locate roots of an equation in an interval.
How to find derivatives of basic functions directly from the definition of a derivative.
How to find velocity, tangent lines and other rates of change by using the derivative.
How to find the derivatives by using the product rule and quotient rule.
KEY CONCEPTS AND MAIN DEFINITIONS:
Function, Graph of a function, Vertical line test, Domain and Range of a function, Natural
domain of a function given by an expression, Composition of f and g, One-to-one functions,
Horizontal Line test, Inverse of a function, natural logarithmic and natural exponential function,
epsilon–delta definition of limit of f(x) as x→a, Multiplying top & bottom by the conjugate to find
limits, Continuous function, Derivative of a function, Velocity of a particle, Slope of tangent line to a
curve, Higher order derivatives, the Product Rule & the Quotient Rule for derivatives.
MAIN FORMULAS AND THEOREMS:
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(f+g) (x) = f(x) + g(x) and dom(f+g) = dom(f)∩dom(g); Also f(f -1(x) = x = f -1(f(x)).
(fog) (x) = f(g(x)) and dom(fog) = dom(g) – all x for which g(x) is not in dom(f).
ex = y if and only if x = ln(y), ln ( ex ) = x, and e ln(x) = x.
sin-1(x) is the unique angle θ, in [–π/2, π/2], such that sin(θ) = x.
limx→0 sin(x)/x = 1, limx→0 [1-cos(x)] / x2 = 1/2, and limx→∞ (1 + a/x)x = ea
limx→a n√{g(x)} = n√{limx→a g(x)} & limx→a f{g(x)} = f{limx→a g(x)} if f is continuous.
Theorem on Limits [limit of f(x)+g(x), f(x)-g(x), f(x).g(x), f(x)/g(x), f(g(x)) as x→ a]
Theorem on Continuity [continuity of f(x)+g(x), f(x)-g(x), f(x).g(x), f(x)/g(x), f(g(x))]
Definition of the derivative: f ’(x) = limh→0 {[f(x+h) –f(x)] / h}. Also (xn)’ = n.xn-1.
The velocity of a particle with position function f(t) is given by v(t) = f ’(t).
{f(x) + g(x)}’ = {f ’(x) + g’(x)}, {f(x) . g(x)}’ = {f ’(x) . g(x)} + {f(x) . g’ (x)} .
{c . f(x)}’ = c .{f ’(x)}, {f(x) / g(x)}’ = [{f ’(x) . g(x)} - {f(x) . g’ (x)}] / [{g(x)}2].