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MATH 100 - Terms
The limit of a function at x = a - correct answer Suppose f(x) is defined when x is near a. Then, we write: [x->a] lim f(x) = L We say: "the limit of f(x), as x approaches a equals L Vertical asymptote definition - correct answer A vertical asymptote occurs when you have denominator equal to zero Rational Function - correct answer A rational function has the form: f(x) = P (x) / Q (x) [Q =/= 0] Type 1 function - correct answer Horizontal Asymptote: y = 0 Type 2 function - correct answer Horizontal Asymptote: y = leading coeffiecent of top / leading coeffiecent of the bottem Type 3 function - correct answer Horizontal Asymptote: DNE (slant asymptote) Continous - correct answer Function is continuous because we don't need to lift our pencils Not continious - correct answer Function is not continuous because we need to lift our pencils Continuity - correct answer A function f is continuous at x = a if:
f(a) is defined [x->a] lim f(x) exists [x->a] lim f(x) = f(a) Continuity from the right / left - correct answer A function f is continuous from the right if: [x->a+] lim f(x) = f(a) from the left if: [x->a-] lim f(x) = f(a) Types of functions continuous at every number in their domains - correct answer Polynomial. Rational, radical, Trigonomertric, inverse trigonometric, exponential, logarithmic Intermediate Value Theorem - correct answer Suppose that f is continuous on the closed interval [a,b]. If f(a) < 0 and f(b) > 0, then there exists a number x = c between x = a and x = b such that f(c) = 0 Tangent line - correct answer Requires only one point and touches the graph only once Secant line - correct answer Requires two points and touches the graph twice Type 1 Tangent Slope - correct answer m = [x->a] lim f(x) - f(a) / x - a Type 2 Tangent Slope - correct answer m = [h->0] lim f(a + h) - f(a) / h Average Rate of Change - correct answer Slope of the secant line:
Linearization of f at x = a - correct answer L(x) = f(a) + f'(a)(x - a) dy - correct answer f'(x) dx delta y - correct answer f(x + delta x) - f(x) Purpose of Newton's Method - correct answer Using tangent lines to approximate the roots of a polynomial equation Newton's Method equation - correct answer x [n + 1] = x [n] - f(x [n]) / f'(x [n]) Rolle's Theorem - correct answer Let f be a function that satisfies the following Three hypothesis: f is continuous on the closed interval [a, b] f is differentiable on the open interval (a, b) f(a) = f(b) Then, there is a number c in (a, b) such that f'(c) = 0 Mean Value Theorem for Derivatives - correct answer Let f be a function that satisfies the following: f is continuous on [a, b] f is differentiable on (a, b) Then, f'(c) = [ f(b) - f(a) ] / [ b - a] If f'(x) > 0 - correct answer f is increasing If f'(x) < 0 - correct answer f is decreasing
If f'(x) = 0 - correct answer f is constant Critical Numbers - correct answer A critical number of a function f is a number c in the domain of f such that: Type 1: f'(c) = 0 Type 2: f'(c) = DNE Local Extrema - correct answer The number f(c) is a: Local maximum value of f if f(c) ≥ f(x) when x is near c Local minimum value of f if f(c) ≤ f(x) when x is near c Extreme Value Theorem - correct answer If f is continuous on a closed interval [a, b] then: f attains an absolute maximum value of f(u) at x = u in [a, b] f attains an absolute minimum value of f(v) at x = v in [a, b] Fermat's Theorem - correct answer If f has a local maximum or minimum at c and if f'(c) exists, then f'(c) = 0 First Derivative Test - correct answer Suppose that c is a critical number of a continuous function f: if f' changes + to - then f has a relative maximum value of f(c) at x = c if f' changes - to + then f has a relative minimum value of f(c) at x = c
- Find the domain of the objective function
- Use first or second derivative test to find the absolute extremum Case 1 of optimization problems - correct answer Domain: [a, b] Absolute min = smallest number in domain Absolute max = Largest number in domain Uses Extreme Value Theorem Case 2 of optimization problems - correct answer Domain: (a, infinity) Uses Critical Point Theorem (First Derivative Test for absolute extrema) Critical Point Theorem - correct answer Suppose f is continuous on I = (a, infinity), and f has exactly one type 1 critical number c in (a, infinity) If f has a relative maximum at c then it is the absolute maximum of f on I If f has a relative minimum at c then it is the absolute minimum of f on I L'Hospital's Rule - correct answer Suppose f and g are differentiable and g'(x) =/= 0 on an open interval I that contains a If [x->a] lim f(x) / g(x) = 0 / 0 or infinity / infinity then [x->a] lim f'(x) / g'(x)
