FORMULAS REVIEW SHEET ANSWERS

1) SURFACE AREA FOR A PARAMETRIC FUNCTION

2) TRAPEZOIDAL APPROXIMATION OF THE AREA UNDER A CURVE (BOTH FORMS) Recall that T n was all about approximating the area under a curve. If you subdivide an interval [ a, b ] into equal sized subintervals, then you can imagine a string of inputs or points c 0, c 1, c 2, c 3, …, c n- 1, c n between a and b, and you can write the trapezoidal sum as L n is the left-hand approximation and R n is the right hand approximation for the area under a curve.

3) THE MACLAUREN SERIES FOR….

4) LIMIT DEFINITION OF THE DERIVATIVE (BOTH FORMS)

5) THE VOLUME OF TWO FUNCTIONS

6) THE COORDINATE WHERE THE POINT OF INFLECTION OCCURS FOR A LOGISTIC FUNCTION If the general form of a logistic is given by then the coordinate of the point of inflection is

7) A HARMONIC SERIES Notice the request was for a harmonic series. There are many and they all diverge:

8) DISPLACEMENT IF GIVEN A VECTOR-VALUED FUNCTION

9) MVT (BOTH FORMS) If a function is continuous, differentiable and integrable, then Think about it, they really are the same formula

10) ARC LENGTH FOR A RECTANGULAR FUNCTION

11) THE DERIVATIVE AND ANTIDERVIATIVE OF LN(AX)

12) LAGRANGE ERROR BOUND

13) THE PRODUCT RULE

14) THE SOLUTION TO THE FOLLOWING DE: DP/DT =.05P(500-P), & IVP: P(0) = 50 See #6 above because that logistic function is the general solution to this specific logistic DE (differential equation) where k = 0.05 & M = 500. Now use the initial condition to find a :

15) VOLUME OF A SINGLE FUNCTION SPUN ‘ROUND Y-AXIS

16) HOOKE’S LAW FUNCTION AND THE GENERAL FORM OF THE INTEGRAL THAT COMPUTES WORK DONE ON A SPRING F ( x ) = kx where k is the spring constant and x is the distance the spring is stretched/compressed as a result of F force can be integrated to get work: where a = initial spring position and b = final spring position.

17) AVERAGE RATE OF CHANGE

18) ALL LOG RULES

19) DISTANCE TRAVELED BY A BODY MOVING ALONG A VECTOR-VALUED FUNCTION

20) A LEAST TWO LIMIT TRUTHS (YOU KNOW AT LEAST EIGHT)

21) CONVERSION FORMULAS: POLAR VS. RECTANGUALR

22) AREA OF A TRAPEZOID

23) IF GIVEN POSITION FUNCTION IN RECTANGULAR FORM: SPEED

24) THE FOLLOWING ANTIDERIVATIVE  ln[ f ( x )] + C

25) VOLUME OF TWO FUNCTIONS (AS ABOVE) SPUN AROUND AN AXIS TO THE LEFT OF THE GIVEN REGION Assuming that the axis is something of the form x = q,

26) THE QUADRATIC THEOREM (NOT JUST THE FORMULA) If given an equation of the form ax 2 + bx + c = 0, then the solutions to this quadratic can be found by using

27) SIMPSON’S RULE FOR THE APPROXIMATION OF THE AREA UNDER THE CURVE If you apply what was said above for the Trapezoidal approximation (#2 above) with an even number of subintervals, then the Simpson’s approximation is given by

28) GENERAL FORMULA FOR A CIRCLE CENTERED ANYWHERE where ( a, b ) is the center

29) TAYLOR’S THEOREM If you want to approximate the value of a function, like sin x, you need some process or formula to do it. Taylor decided that a polynomial could approximate the value of a function if you make sure it has the requisite juicy tidbits: the same value at a center point ( x = a ), the same slope at that point, the same concavity at that point, the same jerk at that point, and so on. The led him to create the following formula: And centered at x = 0 (Maclaurin), He also pointed out that if you truncate the polynomial to n terms, then the part you cut off (the “ tail ” ), R n, represents the error in doing the cutting.

30) THE CHAIN RULE

31) VOLUME OF A SINGLE FUNCTION SPUN ROUND THE X-AXIS

32) ALTERNATING SERIES ERROR BOUND

33) THE THREE PYTHAGOREAN IDENTITIES

34) FIRST DERIVATIVE OF A PARAMETRIC FUNCTION

35) VOLUME OF TWO FUNCTIONS SPUN ‘ROUND AN AXIS THAT IS ABOVE THE GIVEN REGION If y = q is above the function f ( x ), then the volume is given by

36) VOO DOO This is also known as the Integration by Parts process:

37) ARC LENGTH FOR A POLAR FUNCTION

38) FTC (BOTH PARTS) If a function is continuous, then (part I) and if F ( x ) is an antiderivative of f ( x ), then (part II)

39) ANTIDERVIATIVE OF A FUNCTION

40) AVERAGE VALUE OF A FUNCTION

41) THE DE THAT IS SOLVED BY Y=PE^N

42) THE GENERAL LOGISTIC FUNCTION where M = the Max value of the population (or where the population is heading), k = the constant of proportionality, and a = a coefficient found with an initial value.

43) VOLUME OF TWO FUNCTIONS (AS ABOVE) SPUN ‘ROUND AN AXIS THAT IS BELOW THE GIVEN REGION If y = q is below the given region, then the volume is given by

44) AREA OF A EQUILATERAL TRIANGLE IN TERMS OF ITS BASE

45) SECOND DERIVATIVE FOR A PARAMETRIC FUNCTION

46) IF GIVEN A POSITION VECTOR-VALUED FUNCTION: SPEED

47) ARC LENGTH FOR A PARAMETRIC FUNCTION

48) AN ALTERNATING HARMONIC SERIES Again, note that the prompt requests an alternating series. There are many: And all alternating harmonics are convergent by the AST (alternating series test).

49) DERIVATIVE OF THE FOLLOWING FUNCTION Y= B’

50) THE X-COORDINATE OF THE VERTEX OF ANY QUADRATIC FUNCTION

51) MAGNITUDE OF A VECTOR

52) AT LEAST ONE LIMIT EXPRESSION THAT GIVES YOU THE VALUE OF E

53) THE MACLAUREN SERIES FOR SIN(X), COS(X), AND E^X

54) SLOPE OF AN INVERSE FUNCTION AT THE INVERTED COORDINATE

55) THE QUOTIENT RULE

56) SOH-CAH-TOA WITH A RIGHT TRIANGLE DRAWING

57) GENERAL GEOMETRIC SERIES AND ITS SUM

58) SLOPE OF A LINE NORMAL TO A CURVE If m = f’(x) represents the slope of the tangent line to a curve or the instantaneous rate of change of f ( x ), then the slope of the line normal to the curve is given by

59) DISTANCE TRAVELED BY A RECTANGULAR FUNCTION This is the same as arc length:

60) VOLUME OF 2 FUNCTION (AS ABOVE) SPUN ‘ROUND AN AXIS THAT IS TO THE RIGHT OF THE GIVEN REGION If x = q is to the right of the given region, then the volume is given by

61) SURFACE AREA FOR PARAMETRIC FUNCTIONS SPUN ‘ROUND BOTH THE X AND Y- AXES For spinning around the x -axis For spinning around the y -axis:

62) NEWTON’S LAW OF COOLING DE AND GENERAL SOLUTION FUNCTION

63) AREA BETWEEN TWO FUNCTIONS (AS ABOVE)

64) CHANGE OF BASE FOR LOGS (18???)

65) DERIVATIVE FOR AS MANY INVERSE TRIGONOMETRIC FUNCTIONS AS YOU CAN REMEMBER