Enrichment, Extension, and Application

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Exercises

  1. Cotangent Curves

    Two curves which both pass through the point \( P \) are said to be cotangent at \( P \) if they have the same tangent line at \( P \).

    1. For what values of the constants \( a \) and \( b \) are the curves \( C_1 : y = \frac{4}{x} + 2 \), and \( C_2 \): \( y = x^2 + ax + b \) cotangent at the point \( P \) at which \( x = 2 \)?
    2. Make a sketch to illustrate your answer.
  2. Using definitions of continuity, \( \displaystyle \lim_{x \to a} ~ f(x) = f(a) \), and the derivative, \( \displaystyle \lim_{x \to a} ~ \dfrac{f(x) - f(a)}{x - a} \)
    1. Suppose that \( f(x) = x g(x) \) for a function \( g(x) \) which is continuous at \( x=0 \). Explain how you know that \( f^{\prime}(0) \) exists and is equal to \( g(0) \).
    2. An object moves in a straight line with velocity \( v(t) \) given by \[ v(t) = \begin{cases} t^2 & \text{if }t \leq 1 \\ mt+b & \text{if }t > 1 \end{cases} \] where \( m \) and \( b \) are constants. In order to prevent ‘jerk’ in the motion, it is essential that the acceleration of the object, \( a(t) = \dfrac{dv}{dt} \), be continuous for all \( t \geq 0 \). Find values for \( m \) and \( b \) which will guarantee this.
  3. Parabolic tangents and peculiar perpendicularities!
    1. Let \( \ell_a \) and \( \ell_b \) be tangent lines to the parabola \( y = x^2 \) at \( x = a \) and \( x = b \), respectively, where \( a \neq 0 \).
      1. For what values of \( a \) and \( b \) are \( \ell_a \) and \( \ell_b \) perpendicular to one another?
      2. Make a sketch to illustrate one case of your answer, for \( a = 2 \).
    2. The tangent line to the parabola \( x^2 = 4y \) at the point \( P~(a,b) \) meets the line \( y = -1 \) at the point \( A \). Suppose \( B \) is the point \( (0, 1) \).
      1. Find the \( x \)-coordinate of the point \( A \) in terms of \( a \).
      2. Show that for all \( a \neq 0 \), \( \angle ABP \) is a right angle.
  4. Square roots unlimited! Sketch of curve defined implicitly by sqrt(x) + 2*sqrt(y) = 4

    To the right is a sketch of the curve defined implicitly by the equation \( \sqrt{x} + 2\sqrt{y} = 4 \), for \( x \geq 0 \).

    1. Determine \( y \) explicitly as a function of \( x \) for this curve, for \( x \geq 0 \).
    2. Determine the value of the constant \( k \) such that the line \[ y = -x+k \] is tangent to this curve.
  5. No hyperbole here!
    1. Suppose that \( a \gt 0 \). Sketch the first quadrant portion of the curve \( y = \frac{a}{x} \). Pick any point \( P~\left( a, \frac{a}{c} \right) \) on this curve, and sketch the tangent line at \( P \), labelling its \( y \)-intercept \( A \) and its \( x \)-intercept \( B \). Show that \( P \) is the midpoint of the line segment \( AB \), regardless of the position of \( P \).
    2. Let \( O \) represent the origin. Show that the area of the triangle \( AOB \) is a constant, also independent of the position of \( P \).
  6. Relating area derivatives to perimeters, volume derivatives to surface areas.

    We begin by investigating how the area of a square (or circle) varies as its side length (or radius), \( x \), changes.

    Two squares, one of side length x, one of side length x + delta x
    Two squares, one of radius x, one of radius x + delta x

    The area of a square with variable side length \( x \gt 0 \) is \( A_s(x) = x^2 \); for a slightly different side length \( x + \Delta x \), the area is \( A_s(x + \Delta x) = (x + \Delta x)^2 \).

    Similarly, the area of a circle with variable radius \( x \gt 0 \) is \( A_c(x) = \pi x^2 \); for a slightly different side length \( x + \Delta x \), the area is \( A_c(x + \Delta x) = \pi(x + \Delta x)^2 \).

    Note that, for the square, \( \dfrac{dA_s}{dx} = 2x \), which is half the perimeter of the original square, while for the circle, \( \dfrac{dA_c}{dx} = 2\pi x \), which is identically equal to the perimeter of the original circle.

    The geometry behind this is revealed in the rightmost shape in each pair of shapes shown.

    In the case of the square, we see that increasing the side length of the square by \( \Delta x \) adds area \( \Delta A_s \), equal to the area of the grey strips shown on two sides of the square, plus the area of the small black square in the corner, i.e. \( \Delta A_s = 2x \Delta x + (\Delta x)^2 \). The area of the small black square is \( \Delta x ^2\), and we see why only the \( 2x \) term survives when we first find the derivative: \[ \dfrac{dA_s}{dx} = \lim_{\Delta x \to 0} ~ \dfrac{\Delta A_s}{\Delta x} = \lim_{\Delta x \to 0} ~ \dfrac{2x \Delta x + (\Delta x)^2}{\Delta x} = \lim_{\Delta x \to 0} ~ (2x + \Delta x) = 2x \]

    The circle's variation is similar, but less obvious geometrically. In this case, \( \Delta A_c \) equals the area of the grey band of width \( \Delta x \) around the circle, which is \[ \Delta A_c = \pi (x + \Delta x)^2 - \pi x^2 = 2\pi x \Delta x + \pi (\Delta x)^2 \]

    The first term is the area of a strip of length \( 2\pi x \) (the inner circumference of the grey band), and width \( \Delta x \). The second term represents the additional area due to the longer outer circumference \( 2\pi ( x + \Delta x) \) of the grey band, which approaches \( 2\pi x \) as \( \Delta x \to 0 \).

    Thus, the derivative is \( \displaystyle \dfrac{dA}{dx} = \lim_{\Delta x \to 0} ~ \dfrac{dA}{dx} = \lim_{\Delta x \to 0} ~ \dfrac{2 \pi x \Delta x + \pi (\Delta x)^2}{\Delta x} = 2\pi x \).

    1. Discover how this idea generalizes by relating volume derivatives to surface areas.
      1. Show that for a cube with variable side length \( x \gt 0 \), the derivative of the volume, \( V \), is identically equal to half the surface area, \( S \). Make a sketch similar to that for the square to illustrate why this result holds.
      2. Show that for a sphere of variable radius \( x \gt 0 \), the derivative of the volume ,\( V \), is identically to the surface area. Explain the geometry briefly.
    2. Show that, for a pyramid with square base of variable side length \( x \gt 0 \)) and height \( kx \), there is only one value of the constant \( k \) for which the derivative of the volume \( V \) is identically equal to half the surface area \( S \).
    3. For a cylinder of variable radius \( x \gt 0 \) and height \( kx \), find a value of \( k \) which makes the derivative of the volume \( V \) equal to half the surface area \( S \). Then, find a second value of \( k \) such that \( \dfrac{dV}{dx} = S \).
    4. Show that, for a ‘pill capsule’ consisting of hemispherical caps of variable radius \( x \gt 0 \) on a cylinder of length \( kx \), there is no value of \( k \gt 0 \) for which the volume derivative \( \dfrac{dV}{dx} \) is identically equal to half the surface area, \( S \).
    5. For each of the solids in parts b), c), and d), determine all possible values of the constant \( a \) for which there is a positive constant \( k \) such that \( \dfrac{dV}{dx} = aS \).
  7. Odds are, it's even!
    1. Suppose that \( h(x) \) is an odd, differentiable function. Show that \( h^{\prime}(x) \) is an even function.
    2. Suppose that \( g(x) \) is an even, differentiable function. Show that \( g^{\prime}(x) \) is an odd function.
    3. Suppose that the quadratic function \( f(x)=ax^2 + bx + c \) has real, unequal roots \( r \) and \( s \). Show that \( f^{\prime}(r) = - f^{\prime}(s) \).
  8. Touchy cubic tangents...
    1. Make a careful sketch of the cubic curve \( y = x^3 \).
    2. For any \( a \neq 0 \), the tangent line at any point \( P~(a , a^3) \) on this curve will intersect the curve again at a point \( Q~(b, b^3) \). Show that the slope of the curve at \( Q \) is four times its slope at \( P \). Illustrate this by adding a tangent line to your sketch at \( P~(1, 1) \), and the tangent at the corresponding \( Q \).
  9. Fun with functional equations!
    1. Suppose that \( f \) is a function satisfying the equation \( f(x + y) = f(x) + f(y) \) for all real \( x \) and \( y \), and that \( f^{\prime}(0) = c \), a constant.
      1. Show that \( f(0) = 0 \), and hence that \( f^{\prime}(0) = \displaystyle \lim_{h \rightarrow 0} ~ \dfrac{f(h)}{h} \).
      2. Use the definition of the derivative, \( f^{\prime}(x) = \displaystyle \lim_{h \rightarrow 0}~\dfrac{f(x+h)-f(x)}{h} \), and the given equation for \( f(x + y) \), to show that \( f^{\prime}(x) = c \) for all real \( x \), i.e., that the graph of \( y = f(x) \) has slope \( c \) for all real \( x \).
    2. Suppose that \( f \) is a function satisfying two equations: \( f(x + y) = f(x)f(y) \), and \( f(x) = 1 + xg(x) \), where \( \displaystyle \lim_{x \rightarrow 0} ~ g(x) = 1 \). Use the definition of the derivative (as in part a)) to show that \( f^{\prime}(x)=f(x) \) for all real \( x \). What elementary functions could \( f(x) \) be?
    1. Let \( f(x) = x \left\lvert x \right\rvert \). Find \( f^{\prime}(0) \), or show that it does not exist.
    2. Recall that \( \sqrt{x^2} = \left\lvert x \right\rvert \). Let \( f(x) = \left\lvert x \right\rvert \), and show that \( f^{\prime}(x)=\frac{x}{\left\lvert x \right\rvert} \).
    3. Consider the function \( f(x) = \left\lvert \left\lvert x \right\rvert - 1 \right\rvert \).
      1. Sketch the graph of \( y= \left\lvert x \right\rvert - 1 \).
      2. Use your graph in part i) to sketch the graph of \( y = f(x) \).
      3. Find an explicit piecewise definition of \( f(x) \).
      4. Indicate any points where \( f^{\prime}(x) \) does not exist, giving reasons. Find an explicit expression for \( f^{\prime}(x) \) wherever it does exist. (It will be a piecewise-defined derivative function.)
  10. Recall that the squeeze theorem (a.k.a. the sandwich theorem) states that
    If \( m(x) \leq f(x) \leq M(x) \) for all \( x \) near \( x = a \), and \( \displaystyle \lim_{x \rightarrow a}~m(x) = \displaystyle \lim_{x \rightarrow a} ~ M(x) = L \), then \( \displaystyle \lim_{x \rightarrow a} ~ f(x) = L \)
    1. Suppose that a function \( f(x) \) satisfies the inequality \( \left\lvert f(x) \right\rvert \leq \left\lvert x \right\rvert^3 \) for all \( x \) in an interval containing \( 0 \).
      1. Show that \( f(0) = 0 \).
      2. Use the definition of the derivative, \( f^{\prime}(0) = \displaystyle \lim_{x\rightarrow 0} ~ \dfrac{f(x)-f(0)}{x} \) and the squeeze theorem to show that \( f^{\prime}(0) = 0 \).
    2. If the given inequality were of the form \( \left\lvert f(x) \right\rvert \leq \left\lvert x \right\rvert^{\alpha} \), for what real values of \( \alpha \) other than \( 3 \) would your argument in part a) remain valid?
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