Formulas Review Sheet Answers

1. Formulas Review SheetAnswers

2. 1) Surface area for a parametric function

3. 2) Trapezoidal approximation of the area under a curve (both forms) • Recall that Tn 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 c0, c1, c2, c3, …, cn-1, cn between a and b, and you can write the trapezoidal sum as Ln is the left-hand approximation and Rn is the right hand approximation for the area under a curve.

4. 3) the maclauren series for…. • ; • ;

5. 4) Limit definition of the derivative (both forms)

6. 5) The volume of two functions

7. 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

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

9. 8) Displacement if given a vector-valued function

10. 9) MVT (both forms) • If a function is continuous, differentiable and integrable, then Think about it, they really are the same formula

11. 10) Arc length for a rectangular function

12. 11) The derivative and antiderviative of ln(ax)

13. 12) LaGrange error bound 

14. 13) The product rule

15. 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:

16. 15) Volume of a single function spun ‘round y-axis

17. 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.

18. 17) Average rate of change

19. 18) All log rules

20. 19) Distance traveled by a body moving along a vector-valued function

21. 20) A least two limit truths (you know at least eight)

22. 21) Conversion formulas: polar vs. rectangualr

23. 22) Area of a trapezoid

24. 23) If given position function in rectangular form: speed

25. 24) The following antiderivative ln[f (x)] + C

26. 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,

27. 26) The quadratic theorem (not just the formula) If given an equation of the form ax2 + bx + c = 0, then the solutions to this quadratic can be found by using

28. 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

29. 28) General formula for a circle centered anywhere where (a, b) is the center

30. 29) Taylor’s theorem If you want to approximate the value of a function, like sinx, 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”),Rn, represents the error in • doing the cutting.

31. 30) The chain rule

32. 31) Volume of a single function spun round the x-axis

33. 32) Alternating series error bound

34. 33) The three Pythagorean identities

35. 34) First derivative of a parametric function

36. 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

37. 36) Voo doo This is also known as the Integration by Parts process:

38. 37) Arc length for a polar function

39. 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)

40. 39) Antiderviative of a function

41. 40) Average value of a function

42. 41) The de that is solved by y=pe^n

43. 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.

44. 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

45. 44) Area of a equilateral triangle in terms of its base

46. 45) Second derivative for a parametric function

47. 46) If given a position vector-valued function: speed

48. 47) Arc length for a parametric function

49. 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).

50. 49) Derivative of the following function y= b’