Pure Mathematics A4: Real Functions, Graphs, Conics

30 flashcards · Mathematics · by lakesare

1. What domain codomain does f have by convention?

   Domain - all reals for which the domain is applicable Codomain - all reals (ℝ)

2. Where do these asymptotes come from: horizontal asymptote vertical asymptote

   horizontal asymptote - value of y as x tends to ∞. The horizontal asymptote y = a/c arises because, for large positive and negative values of x , the value of the function is approximately ax/cx = a/c.vertical asymptote - y is undefined when denominator is 0. The vertical asymptote x = −d/c arises because the function is undefined when the denominator cx + d is zero.

3. How does the modulus function look?

4. What's the name of the function that looks like this?

   Floor function, or Integer part function f(x) = ⌊x⌋.

5. What are intercepts?

   An intercept is a value of x or y at which the graph y = f (x) of a function f meets the x- o r y -axis, respectively.

6. Increasing on I VS strictly increasing on I?

7. Name 3 curious cases stationary points might represent. Stationary points are values of x at which f '(x) = 0. Note, however, that f ' may do none of these things; for example, a constant function has f '(x) = 0 at all values x in the domain, so every point on its graph is a stationary point.

   local max local min horizontal point of inflection f(x) = x3 has a horizontal point of inflection at x = 0.

8. How can you describe the x = –1 asymptote?

   As x → −1+, f(x) → ∞. As x tends to −1 from the right, f of x tends to infinity.

9. How is "as x → a+" read?

   As x tends to a from the right.

10. What's the dominant term VS coefficient of the dominant term?

    dominant term - anxn coefficient of the dominant term - an

11. What are the construction lines?

12. Why are the construction lines for f(x) are y = x, and y = –x?

    −1 ≤ sin x ≤ 1 −x ≤ f(x) ≤ x So, the graph of f lies between the lines y = x and y = −x.

13. What are the dashed lines?

    These are the construction lines.

14. Why is the sinc function sometimes defined like this?

    Because otherwise it has a hole at x = 0.

15. What's another name for hybrid functions? Hybrid functions have rules defined by different formulas on different parts of their domains.

    Piecewise-defined functions.

16. What's the difference between exponential functions VS hyperbolic functions?

    exponential functions - f(x) = ex the exponential function is the function f(x) = ex where the base e = 2.71828... is Euler's number and the argument x occurs as an exponent. More generally, an exponential function is a function of the form f(x) = abx.hyperbolic functions - f(x) = cosh(x), or f(x) = sinh(x), etc.

17. What does cosh stand for?

    Cosine Hyperbolic

18. Why are these functions called hyperbolic?

    The name 'hyperbolic' originates from the use of these functions as parametric forms for a hyperbola, a type of conic.

19. How are exponential and hyperbolic functions connected?

    By cosh(x) definition!

20. How are hyperbolic functions connected to trigonometric functions?

    cosh(ix) = cos(x)

21. [Conics] How should we place the plane in order to slice the double cone to produce: a circle an ellipse a parabola a hyperbola

    a circle - ⟂ to axis an ellipse - tilted more a parabola - ∥ to side of cone a hyperbola - tilted more

22. [Conics] What's eccentricity (e) equal to algebraically?

    e = distance to focus/distance to directrix

23. [Conics] How does the circle fit into this picture?

    The circle has no focus–directrix property. However, some texts consider the focus to be the centre of the circle, and the directrix to be 'at infinity', so that the circle is obtained when e = 0.

24. [Conics] Non-degenerate conics are not functions. However, how can we still describe a conic using a function?

25. Parametric equations VS parametrisation?

    parametric equations parametrisation A parametrisation of a given curve need not be unique. Different parametrisations of a curve may correspond to different modes of traversing the curve.

26. Name an interesting way to see this scaling.

    y = f(x) And we want to find the relationship between y' and x'. x' = λx; y' = μy. y'/μ = f(x'/λ)

27. What is this?

    A parametrisation.

28. What are the parametric equations of this parametrisation?

29. Name 2 possible parametrisations of the hyperbola.

    1. x = a sec(t); y = b tan(t) 2. x = a cosh(t); y = b sinh(t) Eliminating sec and tan:Eliminating cosh and sinh:

30. Why does the cosh/sinh parametrisation only trace out the right-hand side of the hyperbola?

    Because cosh(t) is always positive. And sinh(t) can be both positive and negative.