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limit y-value that your function appears to approach as x approaches something (a number value) removable discontinuity a missing point end behavior refers to a function's behavior as x → ∞ (eb is positive) or x → -∞ (eb is negative) how to find the limit of a rational algebraic expression (p(x)/q(x)) 1. if the degree of p(x) = the degree of q(x), the limit is the ratio of the leading coefficients 2. if the degree of p(x) < the degree of q(x), the limit = 0 3. if the degree of p(x) > the degree of q(x), simplify the ratio of the leading terms and determine the end behavior of that definition of continuity f(x) is continuous at x = a if: 1. the limit as x approaches a of f(x) exists 2. f(a) exists 3. the limit as x approaches a f(x) = f(a) (basically if you can draw it without lifting your pencil from the page; for every y-value between the 2 end points, there is an x-value that goes with it) discontinuities of rational algebraic functions only 2 types are possible: 1. removable (hole in the graph), which occurs when you cancel a factor 2. vertical asymptote which occurs when you can't cancel a factor f(x) is continuous on [a,b] if 1. f(x) is continuous for all values on (a,b) 2. f(x) is continuous from the right at x = a 3. f(x) is continuous from the left at x = b AROC (Average Rate of Change) slope; f(x₁) - f(x₀)/x₁ - x₀ IROC (Instantaneous Rate of Change) slope of the tangent line at any x-value (limit of AROC) = the derivative of f(x) = f¹(x) = dy/dx; the limit as x₁ → x₀ of f(x₁)-f(x₀)/x₁-x₀ how to find the equation of the tan line to f(x) at a given value of x *need to find 1. point and 2. slope* 1. plug in a value to f(x), get the x- and y-coordinates 2. find IROC 3. write the answers to 1 & 2 in slope-intercept or point-slope formula derivative = IROC = slope of the tangent line = limit of AROC as one point → another; has two formulas tips for graphing f'(x) 1. the slope in your f(x) graph is the y-value in your f'(x) graph 2. at any x-value where f(x) has a horizontal tan line, the y-value in your graph will be 0 3. if the graph of f(x) is increasing on [a,b] then the graph of f'(x) is above the x-axis 4. if the graph of f(x) is decreasing on [a,b] then the graph of f'(x) is below the x-axis Power Rule f(x) = k∙xⁿ → f'(x) = k∙nxⁿ ̄¹ Product Rule if f(x) = g(x)∙h(x) then f'(x) = g(x)∙h'(x) + g'(x)∙h(x) (in a problem, if you're asked to find f'(a): find f'(x) first THEN plug in x = a) Quotient Rule if f(x) = h(x)/g(x) then f'(x) = [g(x)∙h'(x)-g'(x)∙h(x)]/[g(x)]² (DO NOT DO THIS: f'(x) = h'(x)/g'(x) unless you can use L'Hopital's rule) f(x) = sinx fˈ(x) = cosx f(x) = cosx fˈ(x) = -sinx f(x) = tanx fˈ(x) = sec²(x) f(x) = secx fˈ(x) = tanx∙secx f(x) = cotx fˈ(x) = -csc²x f(x) = cscx fˈ(x) = -cotx∙cscx f(x) = logᵨx = lnx fˈ(x) = 1/x f(x) = logᵣx fˈ(x) = 1/xlnr f(x) = eⁿ fˈ(x) = eⁿ bⁿ fˈ(x) = bⁿlnb Chain Rule if f(x) = g(h(x)) then fˈ(x) = gˈ(h(x))∙hˈ(x) how to do implicit differentiation *use when it's too hard to find the derivative the normal way* 1. remove all fractions 2. take d/dx of each piece of the function, treating x as the variable and y as a function of x 3. solve for dy/dx how do to a related rates word problem 1. sketch a picture, label with variables/constants 2. come up with an equation relating the variables (*may have to use given info to either find the value of a variable not explicitly given or relate one variable to another and substitute to eliminate one variable) 3. Identify given and what you're trying to find 4. take d/dt of both sides of the equation 5. plug in values and solve logᵣrⁿ = n lne = logᵨe¹ = 1 log100 = log₁₀10² = 2 e ̄ⁿ∙eⁿ = e ̄ⁿ˖ⁿ = e⁰ = 1 eⁿ∙eⁿ = e²ⁿ L'Hopital's Rule let f(x) = h(x)/g(x) if the lim as x → a, ∞, -∞ of h(x) is 0, ∞, -∞ AND the lim as x → a, ∞, -∞ of g(x) is 0, ∞, -∞ then the lim as x → a, ∞, -∞ of f(x) = lim as → a, ∞, -∞ is hˈ(x)/gˈ(x) (*only use when you get 0/0 or ∓∞/∞, must have a common denominator*) point of inflection point on f(x) where the concavity switches (often at this point, fˈˈ(x) = 0) critical point point on f(x) where fˈ(x) = 0 or DNE relative (local) maximum highest value in its neighborhood relative (local) minimum lowest value in its neighborhood absolute maximum highest y-value in the domain absolute minimum lowest y-value in the domain stationary point point on f(x) where fˈ(x) = 0 (type of critical point) relative extrema occur at critical values or numbers absolute extrema occur at critical values or at endpoints how to find where all relative extrema occur 1st derivative test: 1. take the derivative 2. find critical points, plug in to the number line 3. check intermediate values 4. find absolute max and min 2nd derivative test: 1. take the double derivative 2. plug in the critical point 3. determine the sign of the concavity: if it's positive there is a min at that point, if it's negative there is a max how to find absolute extrema on a closed interval 1. find all critical points on the given interval Option 1: 2. place all critical values and end points of the interval on a number line (*if there's a rel. min here, there MUST be an absolute min because the function does not go back down on the interval) 3. do a test (1st or 2nd derivative) to find where relative extrema are then plug in numbers to f(x) to help resolve any questions Option 2: 2. plug in all values of the number line into f(x) to see which one is biggest and which is smallest 3. determine the largest and smallest of your answers; those are your max and min how to find absolute extrema on an open interval do the exact same process as you would to find the abs extrema on a closed interval but find the LIMITS at the end points instead of plugging the end points back in to f(x), evaluate to determine end behavior