Math 140Exam I Study GuideThe Limit of a Function“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 LLim f(x) = L if and only if:x→aLim 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 limitSqueeze 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→ag(x) = LContinuityIf lim f(x) = f(a), then f is continuous at a x→a1. F(a) is defined2. Lim f(x) existsx→a3. Lim f(x) = f(a)x→aIf 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 0If g is continuous at a, and f is continuous at g(a), then (f*g)(x) – f(g(x)) is continuous at aIf 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=42. Jump – piecewise, absolute value, etc. Right and left limits are not equal.3. Infinite – function goes to positive or negative infinityThese functions are continuous at every number in their domain:- Polynomials- Rational functions- Root functions - Trigonometric functionsMath 140Exam I Study GuideIntermediate Value TheoremSuppose 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) = NProve 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 ChangeThe slope of the tangent line to y=f(x) at the point (a,f(a)) is m = lim f(x) – f(a)/x-a m = lim f(a+h) – f(a)/h x→a h→0Derivative = rate of change = slopeIncreasing function = positive derivativeDecreasing function = negative derivativeWhen is a function not differentiable?1. If f is not continuous at a, then f is not differentiable at a 2. Limit from the left and right are different3. At a cusp4. Vertical tangent If f(x)=c, then f’(x) = 0 Power rule: xn = nxn-1Sum 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“Low D. High – High D. Low, Square the Bottom and off you go.” (bottom*Derivative of the top – top*Derivative of the bottom. Square the bottom)Trig DerivativesSin(x) Cos(x) Cos(x) -Sin(x)Tan(x) Sec2(x) Csc(x) -csc(x)cot(x)Sec(x) Sec(x)tan(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 140Exam I Study GuideImplicit Differentiation: Derivative of y is y’. Differentiate the equation and solve for y’Related Rates:1. Read the question2. Draw a picture (or several)3. Identify variables and constants, and any other information you are given4. Figure out what you want to find, and when5. Find an equation that relates the variable whose derivative you want to find to other variables6. Differentiate the equation with respect to t (time)7. Substitute in known quantities and solve8. Check for