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Limits. F has limit L as x approaches c if for any positive

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1. Calculus AB Notes

3. Properties of Limits

4. Continuity A function is continuous if, for every x in the domain, there is a real value of f(x) A function can be continuous on an interval Function y = f(x) is continuous at an interior point c of its domain if lim(x?c)f(x) = f(c) Function y = f(x) is continuous at: A left endpoint a of its domain if lim(x?a+)f(x) = f(a) A right endpoint b of its domain if lim(x?b-)f(x) = f(b) If f is not continuous at x=c, then c is a point of discontinuity

5. Properties of Continuity If functions f and g are continuous at c, then f+g, f-g, f*g, k*f [for any # k], f/g [if g(c) ? 0], are all continuous at c Composites: if f is continuous at c and g is continuous at f(c), then composite g o f is continuous at c

6. Derivative of a Function Derivative of a function f with respect to variable limh?0 Derivative at a point x=a (instantaneous rate of change at x=a): limx?a Right-hand derivative at a: limh?0+ Left-hand derivative at b: limh?0- Many calculators use a different formula

7. Differentiability Differentiable function: derivative exists at every point of the function’s domain A function is differentiable on a closed interval if it has: A derivative at every interior point of the interval A right-hand derivative at the left endpoint of the interval A left-hand derivative at the right endpoint of the interval Differentiability implies: Local linearity – f resembles its own tangent line when close to the point of differentiation continuity

8. How Derivatives May Fail to Exist at a Point Corner: one-sided derivatives differ Ex.) at x = 0 Cusp: slopes of secant lines approach infinity from one side and negative infinity from the other Ex.) at x = 0 Vertical tangent: slopes of secant lines approach either infinity or negative infinity from both sides Ex.) at x = 0 Discontinuity (removable, jump, etc.)

9. Extreme Value Theorem? If f is continuous on a closed interval [a,b], then there is a maximum and a minimum value of f for the interval (open endpoints and unbound ends are not included)

10. Mean Value Theorem If f is continuous on the closed interval [a,b] and differentiable on interior (a,b), Then there exists at least one point c in (a,b) where i.e. instantaneous rate = average rate of change somewhere in [a,b]

11. Rules for Differentiation Derivative of a constant (c) function: Power rule: n = constant #, and if n<0 the derivative does not exist at x = 0 Constant multiple rule: Sum and difference rule: if u and v are differentiable with respect to x, then their sum and difference are differentiable when v and u are, and

12. Rules for Differentiation (continued) Product rule: Quotient rule: provided v ? 0

13. Extreme Values (Extrema) (f=function, D=domain of f) Absolute extreme values (these are also local extrema): Absolute (global) maximum value on D at c,if and only if f(c) = f(x) for all x in D Absolute (global) minimum value on D at c if and only if f(c) = f(x) for all x in D Relative extrema (local extreme values): (c = interior point in f in D) Local maximum value at c iff f(c) = f(x) for all x in open interval containing c Local minimum value at c iff f(c) = f(x) for all x in an open interval containing c 83211

14. Extrema

15. Increasing/Decreasing/Constant Relations to the derivative f’, if f’ is continuous on [a, b], and differentiable on (a, b): Increasing: if f’>0 at every point in (a, b), then f increases on [a, b] Decreasing: if f’<0 at every point in (a, b), then f decreases on [a, b] Constant: if f’=0 for each point in an interval, then there is a k for which f(x)=k for all x in the interval If f’(x)=g’(x) for each point in an interval, then there is a k where f(x) = [g(x) + k] for all x in the interval

16. First Derivative Test for Local Extrema For continuous function f(x): At a crtical point c: If f’ changes from + to -, then f(x) has a local maximum at c If f’ changes from – to +, then f(x) has a local minimum at c If f’ does not change sign, then f(x) has no local extreme value at c At a left endpoint a: If f’<0 for x>a, f has a local maximum value at a If f’>0 for x>a, f has a local minimum value at a At a right endpoint b: If f’<0 for x<b, f has a local minimum value at b If f’>0 for x>b, f has a local maximum value at b

17. Second Derivative Tests, Concavity On an open interval I, the graph of a differentiable function y=f(x) is: concave up – if y’ is increasing on I if y’’ > 0 on I concave down – if y’ is decreasing on I if y’’ < 0 on I Point of inflection: where the graph of a function has a tangent and concavity changes ? a point where the second derivative is zero

18. Second Derivative Test for Local Extrema If f’(c) = 0 and f’’(c) < 0, then f has a local maximum at x = c If f’(c) = 0 and f’’(c) > 0, then f has a local minimum at x = c

19. End Behavior Models g(x) is a right end behavior model for f if and only if lim(x?8) g(x) is a left end behavior model for f if and only if lim(x?8) g(x) is an end behavior model for f(x) if it is both a right and a left end behavior model The term containing the highest power of x in a polynomial function is the end behavior model for that function In a function that is the quotient of two polynomials, the end behavior model is given by the term containing the highest power in the numerator, divided by the term containing the highest power in the denominator

20. Chain rule If f is differentiable at the value g(x) and g is differentiable at x, Then the composite function (f o g)(x) = f(g(x)) is differentiable at x and (f o g)’(x) = f’(g(x)) * (g’(x))

21. Differentiating Parametrized Curves Parametrized curve: (x(t), y(t)) differentiable at t if x and y are both differentiable at t if all three derivatives exist and , then:

22. Implicit Differentiation Differentiate a function (with x only) with respect to x, using: Then solve for by collecting terms on one side, factoring out and isolating it on one side

23. Separable Differential Equations Separable differential equation: a differential equation that can be expressed as the product of a function of x and a function of y If and , then: Each side can then be evaluated seperately, to find an equation with x and y

24. Derivatives of Trigonometric Functions

25. Derivatives of Inverse Functions Derivatives of inverse functions: if: f is differentiable on every point in an interval I, and on I (i.e. interval cannot have extrema other than endpoints) then: f has an inverse f -1 and is differentiable on every point on I The derivative of a function’s invers is the reciprocal of the original function’s derivative Ex: if: f(1) = 2 and f’(1) = 4 then: f-1(2) = 1 and f’(2) = 1/4

26. Inverse Trigonometric Functions Inverse function – inverse cofunction identities: Calculator conversion identities:

27. Derivatives of Inverse Trigonometric Functions

28. Derivatives of Exponential and Logarithmic Functions

29. Power Rule for Arbitrary Real Powers If u is a differentiable function of x and n is real, Then un is a differentiable function of x and:

30. Solving Optimization Problems Optimize: to maximize or minimize some aspect Pick a variable for the quantity to be maximized or minimized Write a function for the variable, whose extreme value is the information sought Graph / determine reasonable domain Use derivative to find critical points and endpoints, apply to problem

31. Optimization with Economic Applications r(x) = revenue from selling x items, dr/dx = marginal revenue c(x) = cost of producing x items, dc/dx = marginal cost p(x) = r(x) – c(x) = profit from selling x items, dp/dx = marginal profit Maximum profit: if any exist, it occurs at a production level where marginal revenue = marginal cost Minimum average cost: if any exist, it occurs at a production level where average cost = marginal cost When only whole values of the variable are reasonable, the possible values directly above and below the non-whole answer should be checked

32. Related Rates Relationships between rates of change; problem solving: If using t for time, assume all variables are differentiable functions of t Restate given information in terms of chosen variables Find an equation relating the variables Differentiate with respect to t Substitute known values into the differential equation obtained and evaluate

33. Linearization and Newton’s Method Linearization of f at a (if f is differentiable at a): L(x) = f(a) + f’(a) (x-a), an approximation (standard linear approximation of f at a) L(x) is the equation of the tangent to the curve at a Center of the approximation is the point x = a Newton’s Method: Guess a first approximate solution of the equation f(x) = 0 (using graph, etc.) Use first approximation to get successive approximations with:

34. Slope Fields(also called Direction Fields) For , a slope field is a plot of short line segments with slopes f(x,y) on a lattice of points (x,y) in a plane Each segment shows the original function’s slope at that point Slope fields of differential equations can reveal the family of functions that has that derivative (antiderivative) When using slope fields: Horizontal lines show where the derivative = 0; the relationship between x and y at such points can be useful Where the derivative (segment pattern) is positive or negative can be important

35. Antiderivative Antiderivative (sometimes represented by F): g(x) = antiderivative of f(x) if g’(x) = f(x) for all x in the domain There are an infinite number of antiderivatives for a given function These possible antiderivatives, which differ by a constant, are the only antiderivatives of a the function Substitution of a point of the antiderivative may be necessary to find the antiderivative asked for in a problem

36. Riemann Sums - Explanation Let f(x) be a continuous function defined on [a,b]: Partition [a,b] into n subintervals by choosing n-1 points such that: a < x1 < x2 < … < xn-1 < b (x# is a point) a = x0 and b = xn; a partition of [a,b] = P = {x0, x1, x2, … xn} Lengths of intervals are ?x1, ?x2, … ?xn therefore the kth subinterval has a length ?xk A number is chosen from each interval; ck is the number from the kth interval Each subinterval has area ˜ f(ck)* ?xk so f(ck) determines if it is +, -, or 0 Sum of products = a Riemann sum for f on the interval [a,b] = Value depends on partition and the numbers chosen for ck

37. Riemann Sums All Riemann sums for a given interval on a given function converge to a common value if lengths of subintervals tend to zero Norm of a partition = ||P|| = longest subinterval If this tends to zero, then lengths of all subintervals tend to zero

38. Types of Riemann Sums

63. Indeterminate Forms

64. Indeterminate Forms and L’Hôpital’s Rule L’Hôpital’s Rule: (for dealing with 0/0) if f(a) = g(a) = 0, f and g are differentiable on an open interval I containing a and g’(x) ? 0 on I if x ? a (for dealing with 0/0) Or, (for dealing with 8 / 8 , 8 *0, 8 – 8) if f(x) and g(x) both approach 8 as x?a then: This can be used for one-sided intervals as well by choosing I as an open interval with a as an endpoint

65. Other Indeterminate Forms For the indeterminate forms 18, 00, 80, the logarithm of the function can be evaluated using L’Hopital’s Rule and the result can be exponentiated to find the limit of the function: Because: (additionally, a can be finite or infinite)

66. Volumes Volume of a solid with known integrable cross section area: If A(x) = the area of cross sections perpendicular to the x-axis, for a solid from a to b, then the volume from x=a to x=b is: Can integrate with respect to y if equations are in the form x = f(y), integrating from y = a to y = b Cavalieri’s volume theorem: solids of equal height with identical cross sections at each height have equal volume Solids of revolution are formed by revolving a function around a given line; cross sections can be circular Solids with holes can be calculated by finding volume without holes, then subtracting the volume of the hole

67. Volume by Cylindrical Shells Can be used on ‘bundt cake’ shapes Cut cylindrical shells from the inner hole out (vertical cuts perpendicular to base of cake) Cylindrical shells can be rolled out to become rectangular slabs Thickness of each slab is ?x, height is f(x), length is 2 r Region of integration is between the intersections of f(x) with the cake base, so from x=a to x=b: Can be done with respect to y from y = a to y = b if the function is in the form x = f(y) The variable with respect to which it is integrated is the variable of the axis along which the radius changes (base of bundt cake)

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