Enrichment, Extension, and Application

Question 1

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Exercises

  1. Warning! Graphic descriptions!
    1. Given below is the graph of the derivative \( f'(x) \) of a function \( f(x) \) for which \( f(0) = 1 \). Sketch a qualitative graph of \( f(x) \) and give a detailed discussion of the behaviour of \( f(x) \) at each of the points \( x = 1, 2, 3, 4 \). Zero order 1 at x=1,2,3; order 2 at x=4; cusp at x=2;positive end behaviour
    2. Curve is positive on domain t=0 to t_3; max at t=t_2; concave up from 0 to t_1; zero at t=0, t=t_3 To the right is a graph of the altitude \( y(t) \) of a hot air balloon during a flight. Sketch a graph of its vertical velocity \( v = \dfrac{dy}{dt} \) and describe the physical behaviour of the balloon at the times \( t_1\), \(t_2 \), and \( t_3 \). Discuss any differences between the balloon's ascent and descent.
  2. Sunshine and inverse square laws...
    1. The rate \( f(x) \) at which photosynthesis occurs in a leaf depends on the intensity \( x \) of the light on the leaf, according to the function \[ f(x) = \dfrac{Mx}{x + K} - R, x \geq 0 \] where \( M\), \(K \), and \( R \) are positive constants.
      1. Sketch the graph of \( f(x) \) and hence describe how photosynthesis depends on light intensity.
      2. The intensity of light clearly depends on the time of day. Suggest a function \( x(t) \) which might describe this dependence.
      3. What other factors could affect \( f(x) \)?
    2. The illumination, \( I \), by a light source of strength \( S \) at a distance \( d \) is directly proportional to \( S \) and inversely proportional to \( d^2 \). Suppose that one light source of strength \( S_1 \) is placed at \( x = 0\), while another, of strength \( aS_1\), is placed at \( x = x_0 \).
      1. Assuming that the total illumination is the sum of that from the two sources, find an equation for the total illumination \( I(x) \) on an on object at some point \( 0 \lt x \lt x_0 \).
      2. If \( x_0 = 3 \) and \( a = 8 \), sketch the graph of \( I(x) \) and describe how the illumination varies with \( x \).
    3. If a negative charge is placed on the line between two equal positive charges, Coulomb's law states that it will be attracted to each positive charge by a force of magnitude \( \dfrac{k}{d^2} \), where \( k \) is a positive constant.

      Suppose that the positive charges are at \( x = 0 \) and \( x = x_0 \) and the negative charge is at some point \( 0 \lt x \lt x_0 \) (between the two positive charges). Find an equation for the resultant force \( f(x) \) on the negative charge. Sketch a qualitative graph of \( f(x) \) and contrast its behaviour with that of the illumination \( I(x) \).

  3. How much info do you need?
      1. Suppose a certain function satisfies each of the following conditions:
        • \( f(-1) = 0 = f(1) \), \( f(2) = 1 \);
        • \( \displaystyle \lim_{x \to -\infty} ~ f(x) = -1, \lim_{x \to \infty} ~ f(x) = 2 \);
        • \( f'(x) \gt 0 \) on \( \left( -\infty, 0 \right) \) and on \( \left( 1, \infty \right) \);
        • \( f''(x) \gt 0 \) on \( \left( -\infty, 0 \right) \) and on \( \left( 0, 2 \right) \);
        • \( f''(x) \lt 0 \) on \( \left( 2, \infty \right) \);
        Could this function have a vertical asymptote at \( x = 0 \)? A cusp at \( x = 0 \)? A horizontal tangent at \( x = 0 \)?
      2. Sketch the graph of \( y = f(x) \), given the additional conditions:
        • \( f(0) = 1 \);
        • \( f'(x) \lt 0 \) on \( (0, 1) \);
        • \( f'(x) \) is continuous for all real values of \( x \).
    1. Curve through (-2,-2),(0,-5),(2,-2),(3,0);oblique asymptote y=x;y goes to plus/minus infinity, approaches asymptote from above/below, as x goes to plus/minus infinity On the right is the graph of a function \( y = f(x) \).
      1. Sketch the graph of the derivative function \( f'(x) \).
      2. Determine the values of \( x \) which satisfy the inequality \( \left( f(x) \right)^2 \leq x^2 \).
      3. Sketch the graph of the reciprocal function \( g(x) = \dfrac{1}{f(x)} \).
  4. Theoretically speaking...
    1. Use Rolle's theorem to show that if \( a \geq 3 \), then the function \( f(x) = x^3 - ax + b \) cannot have two real roots in the interval \( (-1, 1) \) for any real number \( b \). Illustrate this result with a sketch of the graphs of \( y = f(x) \), for \( a = 1 \) and \( a = 4 \), and using \( b = -1\), \(0 \), and \( 1 \) in each case.
    2. Suppose that \( f(x) \) and \( g(x) \) are differentiable functions such that \( f'(x) \gt g'(x) \) for all real \( x \).
      1. Show that if \( f(a) = g(a) \), then \( f(x) \gt g(x) \) for all \( x \gt a \) and \( f(x) \lt g(x) \) for all \( x \lt a \).
      2. Construct a counterexample which illustrates that the results of part i) do not hold if \( f(a) \neq g(a) \).
    3. Suppose the Tortoise and the Hare start their race at the same time and finish in a tie. Determine which of the following statements must be true, and explain your reasoning.
      1. “At some point during the race, the two were not tied.”
      2. “Their speeds were equal at the end of the race.”
      3. “Their speeds were equal at exactly the same time at some point in the race.”
  5. Asymptotes a' plenty!

    Determine the asymptotic behaviour of each of the given functions below, both as \( x \to \infty \) and \( x \to -\infty \).  Note that some have horizontal asymptotes, some oblique/slanted asymptotes, and some behave asymptotically as power functions.

    Then, find any intercepts, discontinuities, and critical points. Use ONLY the first derivative test to determine a function's behaviour at its critical points (computing the second derivative can be quite time-consuming in some cases).

    Finally, sketch the graph of the function, indicating asymptotes with dashed lines or curves.

    1. \( f(x) = \dfrac{1 + x + x^2 + x^3}{1 + x^3} \) (be careful – what is special about this function?)
    2. \( f(x) = \dfrac{2x^2 + x - 1}{x - 1} \)
    3. \( f(x) = x^2 - \dfrac{2}{x} \)
    4. \( f(x) = x^{\frac{2}{3}} + \dfrac{4}{x^{\frac{2}{3}}} \)
    5. \( f(x) = \left( \dfrac{x + 8}{x} \right) \sqrt{x^2 + 100} \)
  6. To some degree, they must be polynomials!
    1. A certain polynomial \( f(x) \) satisfies the following conditions:
      • Critical points occur at \( x = -1, 1, 2, 3 \)
      • \( f(-1) = 2, f(1) \lt 0, f(2) = 3, f(3) \gt 1 \)
      • \( f''(-1) = 0, f''(1) \gt 0, f''(2) \lt 0, f''(3) = 0 \)
      Sketch the graph of \( f(x) \) as accurately as possible for each of the cases when the degree of \( f \) is even and when it is odd.
    2. Below is the graph of a polynomial \( y = f(x) \). Explain why \( f \) must be a polynomial of at least \( 5^{th} \) degree. Curve with at least 3 inflection points, 3 turning points, at least two zeros of odd order, at least one zero of even order greater than 0
  7. Symmetry rules!
    1. The equation \( y^2 = x^2(2 - x^2) \) can be though to represent two functions: a function \( f(x) \) for \( x \geq 0 \), and a function \( g(x) \) for \( x \lt 0 \). Define the two functions and state their domains. Hint: You will need to recall the definition of \( \sqrt{u^2} \).
    2. Explain why graphing a single quadrant of the curve \( y^2 = x^2(2 - x^2) \) is sufficient to determine the entire curve. Then, sketch the first quadrant portion of \( y = f(x) \).
    3. Does \( f'(0) \) exist? What about \( f'\left( \sqrt{2} \right) \)? Explain your reasoning.
    4. Use your previous results to sketch the entire curve.
  8. Translating cubics for fun and profit!
    1. Find all constants \( a \) such that the cubic polynomial \[ f(x) = x^3 - 3ax^2 + 3x + 1 \] has two distinct relative extremes: a maximum at \( P~(x_1, y_1) \) and a minimum at \( Q~(x_2, y_2) \). Explain why it must be true that \( x_1 \lt x_2 \).
    2. For \( a = 2 \), show that the single point of inflection, \( R(x_3, y_3) \), is at the midpoint of the line \( PQ \).
    3. The result of part b) actually holds in general for all cubics which have both a relative maximum and a relative minimum. However, proving this for a general cubic \[ g(u) = u^3 + Au^2 + Bu + C \] can become pretty involved.

      Instead, we can use the fact that any cubic must have at least one real root, say \( u = r \), to simplify the problem.

      1. Show that the translated function \( f(x) = g(x + r) \) has the form \[ f(x) = x^3 - 3ax^2 + bx \] where \( a, b \) are constants that should be expressed in terms of \( A\), \(B \), and \( r \).
      2. Since \( f(0) = g(r) = 0 \), clearly this translation has shifted the root \( u = r \) to \( x = 0 \). Find conditions on the constants \( a \) and \( b \) such that the remaining two zeros of \( f \), say, \( r_1 \) and \( r_2 \), \(r_1 \lt r_2 \), are real and distinct.
      3. Sketch typical graphs of this cubic for each of \( b \lt 0, b = 0 \), and \( b \gt 0 \), including one for \( a \gt 0 \) and one for \( a \lt 0 \) in each case. Then decide what happens if \( a = 0 \) and sketch the three typical graphs \( (b \lt 0, b = 0, b \gt 0) \) for that case.

        Note: The purpose of this problem is to figure out an approximate location for the roots \( r_1 \) and \( r_2 \), and then sketch rough graphs for the different parameter values.

      4. We know that if \( r_1 \) and \( r_2 \) are real and distinct, then \( f \) will have a maximum, say at \( P~(x_1, y_1) \), and a minimum, say at \( Q~(x_2, y_2) \). Find the values of \( x_1 \) and \( x_2 \).
      5. Show that if \( f(x) \) has a double root at \( x = 0 \), then the other extreme point will occur \( \frac{2}{3} \) as far from \( x = 0 \) as the other root, and the point of inflection will occur \( \frac{1}{3} \) as far.
      6. Find the single point of inflection of the graph of \( f(x) \) in general, and show that if there are two extremes at \( P \) and \( Q \) as above, then the point of inflection lies at the midpoint of the line \( PQ \).
  9. A mystery curve...

    A certain curve has equation \( y^2 + ay = \dfrac{x^2 + b}{cx^2 + dx + e} \), where \( a, b, c, d, e \) are integers.

    This curve has the following properties:

    • its only \( x \)-intercept is at \( (0, 0) \)
    • it has exactly one vertical asymptote at \( x = 1 \)
    • it has two horizontal asymptotes at \( x = -\frac{1}{2} \) and \( x =\frac{1}{2} \)
    1. Determine the values of the integers \( a, b, c, d, e \). Justify your choice.
    2. Sketch the graph of the curve.
  10. Quartic queries… an exploration. Consider the family of quartics which has the form \[ f(x) = x^4 - cx^2 + x \] where \( c \) is an arbitrary real constant.
    1. What tangent line is common to all curves \( y = f(x) \) at the obvious common point, for any value of \( c \)?
    2. Show that there are either no points of inflection on the curve \( y = f(x) \), or there are exactly two points of inflection, symetrically located about some axis.
    3. Considering the possible shapes of the graph of a quartic function, what does the case of no points of inflection infer about the possible number of solutions of \( f(x) = 0 \) (that is, the number of roots of \( f \))? What about the solutions of \( f'(x) = 0 \)? Give a graphical explanation, and sketch a typical graph of \( y = f(x) \) for the case of no inflection points.
    4. Repeat part c) for the case of two points of inflection.

      Hint: Examine how the parameter \( c \) relates to the concavity of the graph \( y = f(x) \) at \( x = 0 \). Think about how this could affect the number of roots of \( f(x) \).

    5. Summarize your results from parts c) and d) with a sketch of several graphs \( y = f(x) \) on the same axes, revealing clearly how the shape of the graph changes as the constant \( c \) varies.
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