Problems

1. Find the derivative of each of the following functions.

   \(y=\sin(2t)-3\cos\,t\)	\(z=\frac{2}{t-1}\)
   \(y=\cos^2 t\)	\(z=\frac{1}{1+7t}\)
   \(y=\cos(t^2)\)	\(y=5^x\)
   \(y=\frac{\sin\,3t}{\cos\,2t}\)	\(y=x^3\sin\,2x\)

2. For each of the following functions,
   1. sketch the graph on paper,
   2. sketch the graph of the derivative on paper.
   You may check your work with the Graph tool.

   \(y=\sin(2t)-3\cos\,t\)	\(z=\frac{2}{t-1}\)
   \(y=\cos^2 t\)	\(z=\frac{1}{1+7t}\)
   \(y=\cos(t^2)\)	\(y=5^x\)
   \(y=\frac{\sin\,3t}{\cos\,2t}\)	\(y=x^3\sin\,2x\)

3. Find the derivative of each of the following functions.

   \(y=\ln(x^{10}-3x)^{1/3}\)	\(y=\ln(e^x)\)
   \(y=\frac{x^2}{1+2x}\)	\(f(x)=3e^{-4x}\)
   \(y=\frac{\cos\,x+x^2}{x^3}\)	\(f(x)=\sin(x^2)\)
   \(y=\frac{\sin\,x}{\cos\,x+1}\)	\(y=y(x)\), where \(y^2+x^2=3x+7\)

4. For each of the following functions,
   1. sketch the graph on papaer,
   2. sketch the graph of the derivative on paper.
   You may check your work with the Graph tool.

   \(y=\ln(x^{10}-3x)^{1/3}\)	\(y=\ln(e^x)\)
   \(y=\frac{x^2}{1+2x}\)	\(f(x)=3e^{-4x}\)
   \(y=\frac{\cos\,x+x^2}{x^3}\)	\(f(x)=\sin(x^2)\)
   \(y=\frac{\sin\,x}{\cos\,x+1}\)	\(y=y(x)\), where \(y^2+x^2=3x+7\)

5. Find all antiderivatives of each of the following functions.

   \(e^{2t}\)	\(\frac{1}{t}\)
   \(t^n\), where \(n\) is a positive integer	\(\sin\,2t\)
   \(\frac{1}{t^2}\)	\(\frac{2}{t-1}\)
   \(\cos\,t\)	\(\frac{1}{1+7t}\)

6. Use the Parametric Plot tool to graph the parametric equations

   \(x=\cos\,t\)  and  \(y=\sin\,t\).

   Explain how the resulting graph relates to Figure 4.
7. Use a trigonometric identity for the cosine of a sum to rewrite the difference quotient

   \(\frac{\cos(t+\Delta t)-\cos\,t}{\Delta t}\)

   as we did for the difference quotient for \(\sin\,t\). Use what you already know about limiting values to show (another way) that

   \(\frac{d}{dt}\,\cos\,t=-\sin\,t\).

8. Find the maximum and minimum values of \(y=\sin\,x+\cos\,x\) on the interval \([0,\pi]\). Explain your reasoning.
9. 1. Show that sine is an odd function and cosine is an even function.
   2. How do the symmetries of these functions fit with the information in Problem 6 in Section 4.5?
10. Give a reason why \(\cos\,x\) is not a polynomial.
    
    From Calculus Problems for a New Century, edited by Robert Fraga, MAA Notes Number 28, 1993.
11. Determine the smallest positive number \(x\) for which the function \(f(x)=-4\sin\left(4x+\frac{\pi}{6}\right)\) has
    1. the value \(0\).
    2. the maximum value of \(f(x)\).
    3. the minimum value of \(f(x)\).
    
    From Calculus Problems for a New Century, edited by Robert Fraga, MAA Notes Number 28, 1993.
12. Recall that the tangent and secant functions are defined by \(\tan\,t=\frac{\sin\,t}{\cos\,t}\) and \(\sec\,t=\frac{1}{\cos\,t}\).

    1. Show that \(\frac{d}{dt}\tan\,t=\sec^2 t\).

    2. Show that \(\frac{d}{dt}\sec\,t=\sec\,t\,\tan\,t\).
    You may find it useful to review Problem 10 in Section 4.7.
13. Recall that the cotangent and cosecant functions are defined by \(\cot\,t=\frac{\cos\,t}{\sin\,t}\) and \(\csc\,t=\frac{1}{\sin\,t}\). Find formulas for the derivatives of

    1. the cotangent function.

    2. the cosecant function.
14. Define a function \(f\) by \(f(x)=\frac{x+\sin\,x}{\cos\,x}\) for \(-\frac{\pi}{2} < x < \frac{\pi}{2}\).
    1. Is \(f(x)\) an even function, an odd function, or neither? Justify your answer.
    2. Find \(f\,'(x)\).
    3. Find an equation of the line tangent to the graph of \(f\) at the point where \(x=0\).
    
    
    Use the Graph tool to check your work.
    From Calculus Problems for a New Century, edited by Robert Fraga, MAA Notes Number 28, 1993.


15. Let \(y=y(x)\) be a function defined implicitly by the equation \(\sin\,x\,\cos\,y=y\). Note that \((0,0)\) satisfies the equation, so, specifically, we take \(y(x)\) to be the solution of the equation that passes through the origin.
    1. Find a formula for \(y\,'\) in terms of \(x\) and \(y\).
    2. With a step size of \(\Delta x=0.1\), use Euler's Method to find approximate values of \(y\) on the interval \([0,1]\), and plot the corresponding points. Repeat with a step size of \(\Delta x=0.01\). Note: Even though we did not start with a differential equation defining \(y\), you created one in part (a), and we have an initial value as well.
    
    
    
    3. Test the accuracy of your Euler solution by substituting \(x=1\) and \(y=y(1)\) (as best you know it) into the equation \(\sin\,x\,\cos\,y=y\). How close is the left-hand side to the right-hand side?

Adapted from "Differentials and Elementary Calculus" by D. F. Bailey, College Mathematics Journal 20 (1989), pp. 52-53.
16. Suppose you walk counterclockwise around the perimeter of the square with corners at \((\pm1,\pm1)\), starting at the point \((1,0)\). Let \((x(t),y(t))\) be your position on the square after you have walked \(t\) units.
    1. Find formulas for \(x(t)\) and \(y(t)\) as functions of \(t\).
    2. Sketch the graph of \(x(t)\) as a function of \(t\).
    3. Sketch the graph of \(y(t)\) as a function of \(t\).
    4. Are the functions \(x(t)\) and \(y(t)\) periodic? If so, what is the period of each? Explain.
       

Problem suggested by John Frampton, Northeastern University.
17. If we begin with \(t_0=0\), then Newton's Method fails to find a root for one of the following functions. Which function - and why?

    (i)    \(f(t)=\sin\,t\)	(ii)    \(f(t)=\cos\,t\)
    (iii)   \(f(t)=2e^t-1\)	(iv)   \(f(t)=e^{-t}-t\)