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Section 3.8 Miscellaneous

Solve the following problems.

  1. For what values of the constant \(c\) does \(\ln x=cx^2\) have solutions. Assume that \(c>0\text{.}\)

  2. Show that \(y=3x^3+2x+12\) has a unique root.

  3. Show that the equation \(x^3+9x+5=0\) has exactly one real solution.

  4. For which values of \(a\) and \(b\) is \((1,6)\) a point of inflection of the curve \(y=x^3+ax^2+bx+1\text{?}\)

  5. Prove that \(\ds f(x)=\frac{1}{(x+1)^2}-2x+\sin x\) has exactly one positive root.

  6. A ball is thrown vertically upwards from a platform 12 feet above the ground so that after \(t\) seconds have elapsed the height \(s\) (in feet) of the ball above the ground is given by

    \begin{equation*} s=12+96t-t^2\text{.} \end{equation*}

    Compute the following quantities:

    1. The initial velocity.

    2. The time to highest point.

    3. The maximum height attained.

  7. The table below gives the values of certain functions at certain points. Calculate each of the following or write “insufficient information” if there is no enough information to compute the value.

    \begin{equation*} \begin{array}{|c|c|c|c|c|} \hline x\amp f(x)\amp f'(x)\amp g(x)\amp g'(x)\\ \hline \hline 1\amp 3\amp 3\amp 2\amp 2\\ \hline 2\amp 4\amp -1\amp 4\amp 0\\ \hline 3\amp 6\amp 1\amp 0\amp 4\\ \hline 4\amp -1\amp 0\amp 1\amp 1\\ \hline 5\amp 2\amp 4\amp 3\amp 3\\ \hline \end{array} \end{equation*}

    1. What is \(\ds \lim _{h\to 0}\frac{f(3+h)-f(3)}{h}\text{?}\)

    2. What is \(\ds \lim _{h\to 0}\frac{f(2+h)g(2+h)-f(2)g(2)}{h}\text{?}\)

    3. Use differentials to find the approximate value of \(f(0.98)\text{.}\)

    4. What are the coordinates of any point on the graph of \(f\) at which there is a critical point?

    1. State domain and range of \(f(x)=\arcsin x\text{.}\)

    2. Derive the differentiation formula \(\ds \frac{d}{dx}[\arcsin x]=\frac{1}{\sqrt{1-x^2}}\text{.}\)

    3. Let \(g(x)=\arcsin (\sin x)\text{.}\) Graph \(g(x)\) and state its domain and range.

    4. For the function \(g(x)\) as defined in part (c) find \(g'(x)\) using any method you like. Simplify your answer completely.

    5. Explain carefully why the equation

      \begin{equation*} 4x-2+\cos \left( \frac{\pi x}{2}\right) =0 \end{equation*}

      has exactly one real root.

  9. Given that

    \begin{equation*} \sinh x=\frac{e^x-e^{-x}}{2} \mbox{ and } \cosh x=\frac{e^x+e^{-x}}{2} \end{equation*}

    1. Find \(\ds \lim_{x\to \infty}\tanh x\text{.}\)

    2. Find the equation of the tangent line to the curve \(y=\cosh x+3x+4\) at the point \((0,5)\text{.}\)

    1. Differentiate \(\ds f(x)=\frac{\ln x}{x}\text{,}\) for \(x>0\text{.}\)

    2. Sketch the graph of \(\ds f(x)=\frac{\ln x}{x}\text{,}\) showing all extrema.

    3. Use what you have done to decide which is larger, \(\ds 99^{101}\) or \(101^{99}\text{.}\)

  11. Answer the following questions. No justification necessary.

    1. What is the general antiderivative of \(f(x)=6x^2+2x+5\text{?}\)

    2. What is the derivative of \(g(x)=\sinh (x)\) with respect to \(x\text{?}\)

    3. If \(f^\prime (x)\) changes from negative to positive at \(c\) the \(f(x)\) has a (pick one)

      1. local maximum at \(c\text{.}\)

      2. local minimum at \(c\text{.}\)

      3. global maximum at \(c\text{.}\)

      4. global minimum at \(c\text{.}\)

    4. If \(x^5+y^5=1\text{,}\) what is \(y^\prime\) in terms of \(x\) and \(y\text{?}\)

    5. If a point has polar coordinates \((r,\theta )=(3,3\pi )\text{,}\) what are its Cartesian coordinates?

    1. State the definition of the derivative of a function \(g\) at a number \(x\text{.}\)

    2. State the Squeeze Theorem, clearly identifying any hypothesis and the conclusion.

    3. State Fermat's Theorem, clearly identifying any hypothesis and the conclusion.

    4. Give an example of a function with one critical point which is also an inflection point.

    5. Give an example of a function that satisfies \(f(-1)=0\text{,}\) \(f(10)=0\text{,}\) and \(f^\prime (x)>0\) for all \(x\) in the domain of \(f^\prime\text{.}\)

    1. State the definition of a critical number of a function \(f\text{.}\)

    2. State the Mean Value Theorem, clearly identifying any hypothesis and the conclusion.

    3. State the Extreme Value Theorem, clearly identifying any hypothesis and the conclusion.

    4. State the definition of an inflection point of a function \(f\text{.}\)

    5. Give an example of a function with a local maximum at which the second derivative is 0.

    6. Give an example of a quadratic function of the form \(f(x)=x^2+bx+c\) whose tangent line is \(y=3x+1\) at the point \((0,1)\text{.}\)

    1. Give an example of a function that is strictly decreasing on its domain.

    2. Give an example of a function \(f\) and an interval \([a,b]\) such that the conclusion of the Mean Value Theorem is not satisfied for \(f\) on this interval.

    3. Use a linear approximation to estimate \(\sqrt{100.4}\text{.}\) Is your estimate larger or smaller than the actual value?

  15. Short definitions, theorems and examples. No part marks given.

    1. Suppose for a function \(y=f(x)\) we have that if \(x_1\not= x_2\) then \(f(x_1)\not= f(x_2)\) for all \(x_1,x_2\) in the domain of \(f\text{.}\) What does this tell you about \(f\text{?}\)

    2. Suppose you need to show that \(\displaystyle \lim_{x\to a}g(x)=L\) using the Squeeze Theorem with given functions \(y=m(x)\) and \(y=n(x)\text{.}\) What condition(s) must the functions \(m\) and \(n\) satisfy?

    3. Identify the theorem that states the following: if any horizontal line \(y=b\) is given between \(y=f(a)\) and \(y=f(c)\text{,}\) then the graph of \(f\) cannot jump over the line; it must intersect \(y=b\) somewhere provided that \(f\) is continuous.

    4. In Figure 3.8.1, which theorem is shown geometrically for a function \(f\) on \([a,b]\text{?}\)

      Figure 3.8.1. Which theorem?

    5. Given a function \(y=f(x)\text{.}\) Suppose that \(f\) has a root in \([a,b]\) and you want to approximate the root using Newton's method with initial value \(x=x_1\text{.}\) Show graphically, how Newton's method could fail.

    6. The graph of the function \(f\) is given in Figure 3.8.2. Clearly graph a possible inverse function for \(f\) over its maximum domain directly on the given Cartesian coordinates.

      Figure 3.8.2. Inverse function?

  16. Draw a graph of a function \(f\) that has an inflection point at \(x=2\) and \(f^\prime(2)\) does not exist.

  17. Draw a graph of a non-linear function \(f\) that satisfies the conclusion of the Mean value Theorem at \(c=-2\) in \([-4,2]\text{.}\)

  18. Let

    \begin{equation*} f(x)=\left\{ \begin{array}{rl} x+4,\amp x\lt 1\\ 5,\amp x=1\\ 2x^2+3,\amp x>1. \end{array} \right. \end{equation*}

    What can you say about differentiability of \(f\) at \(x=1\text{?}\)

  19. Show that \(f(x)=\sinh x\) is an odd function.