Essential Mathematics: Calculus

All Calculus topics from Essential Mathematics. p. 11 - do Osborn's rule later

133 flashcards · Mathematics · by lakesare

1. Who is the co-inventor of calculus together with Newton?

   Gottfried Wilhhelm Leibniz

2. Leibniz notation for derivatives VS Lagrange notation? Joseph-Louis Lagrange was an Italian-French mathematician who Joseph-Louis Lagrange (1736–1813) made important contributions in many areas, including calculus, mechanics, astronomy, probability and number theory. He was appointed as a professor of mathematics at the Royal Artillery School in Turin at the age of only 19, and was a dedicated and prolific mathematician for the rest of his life, working mainly in Turin and Berlin. Gottfried Wil…

   Lagrange notation (or prime notation): Leibniz notation:

3. Why is this the notation for the second derivative?

4. What's a difference quotient?

   Difference quotient is usually the name for the expression above, - which when taken to the limit as h approaches 0 gives the derivative of the function f.

5. Quick, - what's

6. [differentiation rules] What's the product rule?

7. [differentiation rules] What's the quotient rule?

8. [differentiation rules] What's the chain rule?

9. [differentiation rules] What's the inverse rule?

10. [differentiation rules] How to prove all of these rules?

    Product rule - proven from first principles Quotient rule - special case of product & chain rule Chain rule - hard to prove, but makes intuitive sense Inverse function rule - fact that dx/dx = 1, and chain rule

11. [differentiation rules] Express inverse function rule in Lagrange notation.

12. Find derivative of ex from the first principles.

    You can find it from the difference quotient, together with the fact that ex is DEFINED to have (ex)' = 1 at x = 0!

13. Use quotient rule to find that:

14. Chain rule & inverse function rule look like simple multiplication and division. However, that's not what's happening. What is the real interpretation of these rules? As you can see, in Leibniz notation the chain rule looks like an obvious statement about fractions, which makes it easy to remember. You should keep in mind, though, that this is not what it is! The pieces of notation dy/dx, dy/du and du/dx represent derivatives, not fractions. You can see that, as with the chain rule, the Leibniz…

    Chain rule - if y(u) increases 3 units for every u unit, and u(x) increases 2 units for every x unit, - you'd expect y to increase 3 × 2 = 6 units for every x unit. Suppose that y is a function of u, where u is a function of x, as in the statement of the chain rule. Suppose that for a particular value of x, the value of du/dx is 2, and for the value of u that corresponds to this particular value of x, the value of dy/du is 3. Since y is increasing at the rate of 3 units for every unit that u in…

15. What's antiderivatives VS indefinite integrals?

    Antiderivative - any specific function whose derivative is f. F(x) = x2 + 5Indefinite integral - general function (+ c) that describes the complete family of antiderivatives of f. F(x) = x2 + c

16. Imagine some antiderivatives of this graph.

17. Let f be the function with rule f(x) = x5. Find the indefinite integral of f. Hence write down 3 different antiderivatives of f.

18. What's the indefinite integral of xn (when n ≠ -1)?

19. What is the derivative of xn?

    nxn−1

20. Find the indefinite integral of f(x).

21. [Using integration to deduce displacement from acceleration] Assume that an object falling from rest has a constant acceleration of −9.8ms−2, where the positive direction is upwards. Let s be the displacement in metres of the object from its initial position, and let t be the time in seconds since it began falling. Find an equation that expresses s in terms of t.

22. Why does f have to be continuous to have an indefinite integral?

    If f is a function that has an antiderivative, but which isn’t continuous, then the general function F(x) = (formula for any particular antiderivative of f) + cwhere c represents any constant, and whose domain is the same as the domain of f, usually doesn’t describe the complete family of antiderivatives of f. The graphs of(a) f(x)=1/x2(b) the antiderivative F(x) = −1/x of f(x)=1/x2(c) another antiderivative of f(x)=1/x2

23. Prove that all antiderivatives of f(x) = 2x change by the same amount as x changes form x = 1 to x = 2.

    F(x) = x2 + c F(2) - F(1) = 22 + c - 12 - c = 3 The important fact to appreciate is that the amount of the change is the same for all the antiderivatives of f. That’s because the graphs of all the antiderivatives are vertical translations of each other, and translating a graph vertically doesn’t affect how much it changes from one value of x to another.

24. Where does +c come from in the indefinite integral?

25. f(x) has so many antiderivatives! Why is F(b) − F(a) is the same for every antiderivative F of f.

    So if you want to work out the amount by which an antiderivative of a continuous function f changes from x = a to x = b, then you just need to find any antiderivative F of f, and calculate F(b) − F(a).

27. We know that: Why does this formula not give you the antiderivative of 1/x?

    Because ln(x) only applies when x ∈ (0, ∞), but 1/x has domain x ∈ (-∞, 0) ∪ (0, ∞).

28. Find the indefinite integral:

29. Find the indefinite integral:

30. Find the indefinite integral:

31. Find the indefinite integral:

32. [Integration by substitution] Find the indefinite integral.

33. [Integration by substitution] Find the indefinite integral.

34. [ Integration by substitution] Find the indefinite integral.

35. [Choosing the 'something' when we integrate by substitution] Find the indefinite integral.

36. [Integration by substitution]

37. [Integrating a function of a linear expression] Find the integral.

38. How do we determine this?

    Via integration by substitution.

39. In what form should the integrand be to use integration by substitution?

    Set the something equal to u, and find du/dx.

40. In what form should the integrand be to use integration by parts?

    where f is a function that becomes simpler when differentiated, g is a function that you can integrate

41. Which graph shows: derivative, second derivative of the function in graph A?

    derivative of A - C second derivative of A - D (There are various ways to work this out. For example, notice that the function in graph A has a stationary point when x is approximately 0. So the graph of its derivative must cross or touch the x-axis when x is approximately 0, which tells you that the derivative must be the function in graph C. Alternatively, notice that the function in graph A is decreasing for values of x just less than 0. So its derivative must be negative for these values of…

42. Where does the word 'integral' come from?

    'To integrate' means 'to join together'.

43. [Understanding signed areas] What's the signed area from x = −3 to x = 1? The signed area of the region is its area with a plus or minus sign according to whether it lies above or below the x-axis, respectively.

    −2.2 + 4.8 = 2.6.

44. [Signed area] Find the signed area from 3 to 3. Signed area: If there's no mention of 'from x = a to x = b', then to decide whether the signed area is positive or negative you just need to consider whether it lies above or below the x-axis.However, if the text is of the form 'the signed area from x = a to x = b', then to decide whether the signed area is positive or negative you also have to consider whether b is greater than or less than a.

    0.

45. [Signed area] Find the signed area from 6 to 10. Signed area: If there's no mention of 'from x = a to x = b', then to decide whether the signed area is positive or negative you just need to consider whether it lies above or below the x-axis.However, if the text is of the form 'the signed area from x = a to x = b', then to decide whether the signed area is positive or negative you also have to consider whether b is greater than or less than a.

    -5.4 From 6 to 10 is forward.

46. [Signed area] Find the signed area from 6 to -1. Signed area: If there's no mention of 'from x = a to x = b', then to decide whether the signed area is positive or negative you just need to consider whether it lies above or below the x-axis.However, if the text is of the form 'the signed area from x = a to x = b', then to decide whether the signed area is positive or negative you also have to consider whether b is greater than or less than a.

    -1.6 The signed area from x = −1 to x = 6 is −2.4 + 4.0 = 1.6, so the signed area from x = 6 to x = −1 is −1.6.From 6 to −1 is backward.

47. [Signed area] How is the signed area between the graph of f(x) and the x-axis from x = a to x = b denoted? This area is called definite integral.

    This notation is read as 'the integral from a to b of f of x, d x'. The numbers a and b are called the lower and upper limits of integration, respectively.

48. [Understanding definite integrals] What's the value of this definite integral? Hint: notice that in some of these definite integrals the upper limit of integration is less than the lower limit of integration.

49. [Understanding definite integrals] What's the value of this definite integral? Hint: notice that in some of these definite integrals the upper limit of integration is less than the lower limit of integration.

50. [Understanding definite integrals] What's the value of this definite integral? Hint: notice that in some of these definite integrals the upper limit of integration is less than the lower limit of integration.

51. [Using a standard property of definite integrals]

52. Explain why the signed area from a to b will be negative.

    Because signed area is defined as f(endpoint_nearest_a) × subinterval, and each f(endpoint_nearest_a) is negative here.

53. Name 3 ways to find the value of the definite integral.

    Taking a limit If f is a particular continuous function, and a and b are numbers in its domain, then you can sometimes calculate an exact value for the integral by taking a limit.In some cases you might even be able to obtain a general formula for the integral, in terms of a and b.However, this is usually a complicated and difficult process, so normally we don't do it.Approximating using subintervals The first option is to obtain an approximate value by using subintervals in the way that you’ve…

54. What does the integration symbol stand for?

    S - Sum.

55. Write down the limit definition of the definite integral from a to b.

56. [Signed-area-so-far function] Write down the rule of A(x). A(x) = signed area between the graph of f(x) and the x-axis, from s to x.

    Here the variable t has been used in the definite integral instead of the usual variable x, to avoid confusion with the input variable x of the function A

57. [Signed-area-so-far function] Find A(1), A(2) and A(3) given f is the constant function f(x) = 3, and the starting point s = 1 Now let's consider how the value of a signed-area-so-far function changes as the value of the input variable x changes. Let's start by looking at what happens when the function f is a constant function.

    A(1) = 0 A(2) = 3 A(3) = 6

58. [Signed-area-so-far function] How much does A(x) change for each unit by which x changes?

    1.5 square units for each unit by which x changes.

59. [Signed-area-so-far function] What's the APPROXIMATE rate of change of A(x) at x=5?

    3 square units for every unit by which x increases. As we go from 'for every unit' to e.g. 'for every 0.1 unit' - the '3 square units' estimate becomes more and more precise! 💖 In other words, the derivative of the function A is the function f, and so the function A is an antiderivative of the function f.

60. [Signed-area-so-far function] At any value of x, the instantaneous rate of change of A(x) is f(x). What does this fact basically mean?

    It means that A'(x) = f(x). And so the function A is an antiderivative of the function f.

61. [Signed-area-so-far function] Explain why this is true (if A(x) is a signed-area-so-far function).

62. How is the integral defined?

    As the area under the function! The fact that integrals have ANYTHING to do with antiderivatives is a consequence of the fundamental theorem of calculus - it has nothing to do with their definition.

63. State the Fundamental Theorem Of Calculus IN WORDS.

    Signed area under f(x) from a to b is connected to the antiderivative of f(x).

64. Why is the Fundamental Theorem Of Calculus true?

    Because rate of change of A(x) (signed-area-so-far) is f(x) at every x. At any value of x, the instantaneous rate of change of A(x) is f(x).

65. [Evaluating definite integrals] What's the area under the f(x)?

66. What can you say about the areas in a) and b)?

    The area in b) is twice the area in a).

67. What can you say about the areas in a), b) and c)?

    The area in (c) = areas in (a) + area in (b). Remember that the sum rule for definite integrals also holds if you replace the plus sign on each side with a minus sign, because, as mentioned earlier, this is always the case when a constant multiple rule and a sum rule hold.

68. [Integration by substitution] How would you show that: Where u(x) is a function of x.

69. [Evaluating a definite integral by substitution (second method)] Evaluate

70. [Expressing a further factor in terms of u when integrating by substitution] Find the integral

71. [Integrating functions of linear expressions efficiently] Find the following integral efficiently.

    You can find it using integration by substitution.

72. [Integrating functions of linear expressions efficiently] Find the following integral efficiently.

    You can find it using integration by substitution.

73. [Integration by substitution] Name 2 ways to solve this integral.

74. [Integration by parts] Find the following integral.

75. State the integration by parts formula.

76. [Proving the integration by parts formula] Suppose that f and g are functions, and function G is an antiderivative of the function g. Prove the integration by parts formula by solving the following derivative.

77. What method of integration would you choose to find this integral?

    Integration by substitution u = 9x4.

78. What method of integration would you choose to find this integral?

    Integration by parts.

79. What method of integration would you choose to find this integral?

    Using trigonometric identity sin(2θ) = 2 sin(θ) cos(θ).

80. What's the difference between integration by substitution VS integration by parts?

81. How is the chain rule and integration by substitution connected?

    Same process in different directions! d y(u(x)) / dx = dy/du * du/dx

82. [Finding a more complicated integral by substitution] How would you solve this integral by substitution?

    Use the substitution u = x - 1, AND express the other factor (x) in terms of u.

83. What's the definition of indefinite integral VS definite integral?

    indefinite integral - collection of all antiderivatives of f(x). definite integral - signed area between the graph of f(x) and the x-axis from x = a to x = b.

84. [Differential equations] What's a differential equation?

    Equation that includes some unknown function and its derivatives.

85. [Differential equations] What's a dependent & independent variable in these differential equations?

    x - independent (because we differentiate with respect to it) y - dependent

86. [Differential equations] What's the order of differential equations?

    The highest-order derivative present in this equation. Differential equation of first order (since the highest derivative that appears is dy/dx, and differential equation). Differential equation of second order (since the highest derivative that appears is d2y/dx2) The nth derivative of a function is said to be the derivative of order n, and the order of a differential equation is the order of the highest derivative that appears in the equation.

87. [Differential equations] What's considered to be a solution of a differential equation?

    A function y = y(x) that satisfies the differential equation, e.g. y = e2x. Usually a differential equation has more than one solution.The general solution of equation (3) is y = Ae2x , where A is an arbitrary constant.

88. [Differential equations: direct integration] Which of these equations are directly integrable?

    a) - yes b) - no c) - no

89. [Differential equations: separation of variables] Which of these equations are separable?

    a - separable b - separable d - NOT e - separable

90. [Differential equations] Name types of differential equations corresponding to the following methods of solving them: Direct integration Separation of variables

    Direct integration - directly integrable Separation of variables - separable

91. [Differential equations: separation of variables] How would you find the general solution of this differential equation?

    Move y-s to the left, and x-s to the right And integrate both sides with respect to x

92. [Differential equations: separation of variables] What 3 things can happen after you solve the differential equation (by the separation of variables)?

    explicit form implicit form can't integrate (or hard to integrate)

93. What's the equation for 'first quantity is proportional to the second quantity'?

    Where K is a constant

94. [Differential equations: verify the solution] How would you show that a particular equation in the implicit form is the solution to the differential equation? Solution in the implicit form. Example of a solution in the explicit form - in this case we can simply substitute the equation.

    Our goal is to derive dy/dx! To achieve that: Take d/dx from both sides And apply the chain rule whenever needed

95. [Differential equations: practice] What is the modelling assumption behind population growth?

    K is the proportionate growth rate for the population, it is a measure of how fast the population size is increasing or decreasing relative to its size.t - time.P - population size.

96. [Differential equations: practice] This is the population growth model. What's K, P and P0?

    K - proportionate growth rate for the population It is a measure of how fast the population size is increasing or decreasing relative to its size.P - population size at any moment P0 - initial population size

97. [Differential equations: practice] How would you find P(t) - equation for the population size depending on time? Population growth model.

    It's a classic separable differential equation, and it's separable! Move P to the left Integrate with respect to t

98. [Differential equations: practice] How did we derive P(t)?

    Via the modelling assumption 'rate of change of population size is proportional to the population size'. Followed by solving this differential equation. Population growth model.

99. [Differential equations: practice] What can you say about the factor of the population increase over a period of time?

    It only depends on the length of the time interval - not when it started! For example, if the population size increases by a factor of 1.05 (that is, grows by 5%) over some time interval of length one year, say, then it will increase by the factor 1.05 over every time interval of length one year.

100. [Differential equations: practice] What's the factor by which population size increases over time interval T?

     eKT - doesn't depend on P0, or on when that time interval started. K - proportionate growth rate constantT - a given time interval For example, if the population size increases by a factor of 1.05 (that is, grows by 5%) over some time interval of length one year, say, then it will increase by the factor 1.05 over every time interval of length one year.