Guidepost 5 Study Guide Notes

CL3, AL1, AL2, AL3, AL4, CD1, AD1, AD2, AD3 CL3: Calculate a lim using the squeeze Theorem AL1: Use lims to evaluate continuity of a Funcion AL2: JustHry a vertical asymptote using the def. AL3: Justify a horizontal asymptote using the det. ALU: Make a calculation using IVT and explain CD1: use the lim definition to find the derivative of a function (as a funcion) AD1: Interpret the value of a derivative in the context of a problem AD2: Use derivatives to determine intervals of increasing/decreasing and explain. CL3: It there exist funcions g(x) = f(x) ≤n(x) for all x near a Lexcept possibly at w, and lim g(x) = lim n(x) = L then, lim F(X) = L xa xa DEX 1) find the value of lims g(x) and h(x) 7) If both lims are the same, conclude lim f(x) is also that value. xa সে? evaluate: lim x² sin -14 sin()≤1 -x² ≤ x2sin (x) ≤1 lim (-x2) = 0, and lim (x2)=0 x→0 so, lim x2 sin()=0 QEX AL2: 1) Justifying a VA: a function F(X) has a VA at x=d it: lims FCX) = ± X and lim F(X) = ± 0 xa+ xa this means a x approaches a, the function grows infinitely large (0) or infinitely small(-4) ・small positive numbers become large positive numbers -small negative numbers become large negание numbers If one (right or left-hand) lim approaches ±10, there is a VA at x=a AL3: a function (x) has an HA ary = L IF: lim F(X) = L or lim H(X) = L X-M this means that as x moved toward & or -0, (x) approaches a constant value L AL4: It a function (X) is continulty on a closed Interval [a,b], and it N is any number between Ha) and (b), then there exists some point o in ca, b) such that: F(C) = N It you araw a continuous chrue for F(a) to (b), then the function must pass through every value between Ha) and (b) at least once. 1) check continually using famous functions 2) evaluate f(x) at the end points like [0,1], compute (0) and (1) 3) check for opposite signs cone + and one-) CD1: the derivative of a function is defined as: f'(x) = lim F(x+1)-FLX) no η this Formula gives the instantaneous rate of change of f(x) at any point X. It is derived From the concept of the difference quotient which gives the slope of the sec line between 2 points on a function, and then takes the lim as the points get infinitely close. also, F'(a) = lim f(x) - (a) x-a taking the lim as x goes to a tangent line's slope at x = a gives the AD1: the derivative f(x) measures the ROC ofa function at a specified pant, the derivative can represent: velocity. If SCE) is the position of an object, then S'(t) (the derivative) represent ito velocity. growth rate: If P(+) is a population, then P(+) represents the rate at which the population changes cost/revenue change: It CCX) is the cost of producing x items, the C'(x) represents the marginal cost, or the cost of producing one additional unit AD2: a function f(x) 18: increasing where f' * (x) > 0 (positive derivate = upward stope) decreasing where f'(x) 0 concave down where f^ prime prime (X) < 0 interaction points where (p')' * (x) = 0 and the concavity changer compute F'(X) 2) save 7^ prime prime (x) = 0 3) test values in intervals tor concavity.