Some Calculus Theorems
& Graphing Functions

  1. Estimating, based on the graph, what are the values of the following?

    \(f(-2)\)
    \(f'(-2)\)
    \(f''(-2)\)
    \(f'(0)\)
    \(f(2)\)
    \(f'(2)\)
    \(f(3)\)
    \(f'(3)\)

  2. CU Boulder Consider the function \(f(x) = x(x-4)^3.\) Without referring to technology, first verify that these are the first two derivatives of \(f.\) \[ f'(x) = 4(x-4)^2(x-1) \qquad \qquad f''(x)= 12(x-4)(x-2) \] Then answer these questions about the graph \(y = f(x)\)

    1. What are the coordinates of the \(x\)- and \(y\)-intercepts?
    2. On what intervals is the function increasing? Decreasing?
    3. What are the \((x,y)\) coordinates of any local minimum/maximum values?
    4. On what intervals is the graph concave up? Concave down?
    5. Find the \((x,y)\) coordinates of any inflection points, if any exist.
    6. What are the values of the following limits? \[ \lim\limits_{x \to \infty} f(x) \qquad \qquad \lim\limits_{x \to -\infty} f(x) \]
    7. Using everything you now know, accurately sketch the graph \(y = f(x)\)
  3. What is the value of the following limit? \[\lim_{x \to \infty} \frac{7x^7-7}{10+9x+8x^2+7x^3+6x^4+5x^5+4x^6+3x^7} \]
  4. The Extreme Value Theorem guarantees that the function \(f\) defined as \({f(x) = 5x^3-2x^2-2x}\) achieves an absolute (global) minimum value and maximum value on the domain \(\bigl[-\tfrac{1}{2}, 1\bigr]\). Calculate this absolute minimum value and absolute maximum value, and briefly argue why these must be the minimum and maximum.
  5. Since sine is continuous between \(\theta = 0\) and \(\theta = \tfrac{\pi}{2},\) by the Intermediate Value Theorem there must be some number \(c\) between \(0\) and \(\tfrac{\pi}{2}\) such that the output \(\sin(c)\) is equal to the average value of sine on that interval. Find the average value of sine between \(\theta = 0\) and \(\theta = \tfrac{\pi}{2}\), and find such a number \(c\).
  6. Suppose that \(g\) is a function such that \(g(-1) = 1\) and \(g'(x) \geq 3\) for all \(x.\) What is the smallest possible value that \(g(3)\) could be?
  7. Suppose that \(g\) is a function such that \(g(1)=3\) and \(1 \leq g'(x) \leq 4\) for all \(x.\) Show that \(8 \leq g(6) \leq 23.\)

  8. For each of the following statements, decide if it is true or false. If it is false, sketch the graph of a function that serves as a counterexample.

    1. If \(f\) is continuous on the interval \([a,b]\), then there must be some \(c\) in the interval \([a,b]\) such that \(f(c)\) is a maximum on \([a,b]\).
    2. If \(f\) is continuous on the interval \([a,b]\), then there must be some \(c\) in the interval \([a,b]\) such that \(f(c)\) is a minimum on \([a,b]\).
    3. If \(f\) is continuous on the open interval \((a,b)\), then there must be some \(c\) in the interval \((a,b)\) such that \(f(c)\) is a maximum on \((a,b)\).
    4. If \(f\) is defined (not necessarily continuous) everywhere on the interval \([a,b]\), then there must be some \(c\) in the interval \([a,b]\) such that \(f(c)\) is a maximum on \([a,b]\).
    5. If \(f\) has a local minimum or a local maximum at \(x=c\) and if \(f'(c)\) exists, then \(f'(c) = 0\).
    6. If \(f\) has a local minimum or a local maximum at \(x=c\) then \(f'(c) = 0\).
    7. If \(f'(c) = 0\) then \(f\) has a local minimum or a local maximum at \(x=c\)
    8. A functions with domain of all real numbers must have either a local minimum or a local maximum somewhere on it's domain.
    9. If \(f\) is a continuous the interval \([a,b]\) and differentiable on the interval \((a,b)\) then there must be some \(c\) in the interval \((a,b)\) such that \({f'(c) = \frac{f(b)-f(a)}{b-a}.}\)
    10. If \(f\) is a continuous the interval \([a,b]\), then there must be some \(c\) between \(a\) and \(b\) such that \(f'(c) = \frac{f(b)-f(a)}{b-a}.\)
    11. If \(f\) is a polynomial function with root \(r\) then \(f'(r) \neq 0.\)
    12. For a function \(f\) that is differentiable on some domain containing \(a\) and \(b,\) if \(f(a) \lt f(b)\) then \(f'(a) \lt f'(b).\)
    13. For a differentiable function \(f\) that is unbounded above the derivative \(f'\) will also be unbounded above.
  9. James Stewart Show that the following curve has three inflections points and that they all lie on a common line. \[y = \frac{1+x}{1+x^2}\]
  10. What’s an equation for the unique quadratic polynomial function that has a global maximum at the point \((-1,2)\) and passes through the point \((3,4)?\)
  11. What’s an equation for the unique quadratic polynomial function \(f\) such that \(f(3) = f'(3) = f''(3) = 3?\)
  12. What’s an equation for the unique quadratic polynomial function that is decreasing for \(-\infty \lt x \lt 5\) and has a root at \(x = 2?\)
  13. What’s an equation for the unique cubic polynomial function that has local extrema at the points \((1,1)\) and \((3,3)\) and a \(y\)-intercept at \(2?\)
  14. A quadratic polynomial always has a local extrema. A cubic polynomial however may or may not have a local extrema. For a cubic polynomial \({ax^3+bx^2+cx+d}\) devise a quick test in terms of the coefficients \(a,\) \(b,\) \(c,\) and \(d\) to determine whether or not it has a local extrema.
  15. James Stewart Find a function \(f\) such that \(f'(-1) = \tfrac{1}{2} \), \(f'(0) = 0\), and \(f''(x) \gt 0\) for all \(x\), or prove that such a function cannot exist.
  16. Sketch the graph of a differentiable function that has an inflection point at \(x = 2\) but is strictly increasing for all \(x\) around \(2.\)

  17. Without appealing to technology write out an argument that the function \(f(x) = 3x-2\sin(x)+7\) has exactly one zero.