The limit of x approaches a equals L means that as the value of x gets closer to a but not equal to a from either side of a the value of f x gets closer and closer to L Math 140 Exam I Study Guide The Limit of a Function Lim f x L if and only if x a Lim f x L the limit as x approaches a from the right x a Lim f x L the limit as x approaches a from the left x a non zero zero infinite limit Squeeze Theorem If f x g x h x when x is near a except possibly at a and lim f x lim h x L then lim x a x a x a g x L Continuity If lim f x f a then f is continuous at a x a 1 F a is defined 2 Lim f x exists 3 Lim f x f a x a x a These functions are continuous at every number in their domain Polynomials Rational functions Root functions Trigonometric functions If f and g are continuous at a and c is a constant the these are also continuous at a F g f g fg f g if g a is not 0 If g is continuous at a and f is continuous at g a then f g x f g x is continuous at a If f x is not continuous at a then f has a discontinuity at a Types of Discontinuities 1 Removable can be removed by redefining f at one point Usually when a 0 in the denominator divides out with something in the numerator F x x 4 x 7 x 4 Removable discontinuity at x 4 2 Jump piecewise absolute value etc Right and left limits are not equal 3 Infinite function goes to positive or negative infinity Math 140 Exam I Study Guide Intermediate Value Theorem Suppose that f is continuous on the closed interval a b and that f a is not equal to f b Let N be any number between f a and f b then there exists a number c in the open interval a b such that f c N Prove that a function has a root If f is continuous on a b and f a 0 and f b 0 or vice versa then intermediate value theorem tells us that there is some c in a b such that f c 0 Derivatives and Rates of Change The slope of the tangent line to y f x at the point a f a is m lim f x f a x a x a m lim f a h f a h h 0 Derivative rate of change slope Increasing function positive derivative Decreasing function negative derivative When is a function not differentiable If f is not continuous at a then f is not differentiable at a 1 2 Limit from the left and right are different 3 At a cusp 4 Vertical tangent If f x c then f x 0 Power rule xn nxn 1 Sum rule f x g x f x g x Product rule f x g x f x g x g x f x Quotient rule f g gf fg g2 top Derivative of Low D High High D Low Square the Bottom and off you go bottom Derivative of the top the bottom Square the bottom Sin x Tan x Sec x Cos x Sec2 x Sec x tan x Trig Derivatives Cos x Csc x Cot x Sin x csc x cot x csc2 x Chain rule if h x g f x then h x g f x f x if g f x and f x exist the derivative of the outside time the derivative of the inside Math 140 Exam I Study Guide Implicit Differentiation Derivative of y is y Differentiate the equation and solve for y Related Rates 1 Read the question 2 Draw a picture or several 3 4 Figure out what you want to find and when 5 Find an equation that relates the variable whose derivative you want to find to other Identify variables and constants and any other information you are given variables 6 Differentiate the equation with respect to t time 7 Substitute in known quantities and solve 8 Check for reasonableness