2.9: Extra Problems for Chapter 2
- Page ID
- 155809
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)| 1. | Find the derivative of \(f(x) = 4x^{3} - 2x + 1\). |
| 2. | Find the derivative of \(f(t) = 1/\sqrt{2t - 3}\). |
| 3. | Find the slope of the curve \(y = x(2x + 4)\) at the point \((1, 6)\). |
| 4. | A particle moves according to the equation \(y = 1/(t^{2} - 4)\). Find the velocity as a function of \(t\). |
| 5. | Given \(y = 1/x^{3}\), express \(\Delta y\) and \(dy\) as functions of \(\Delta x\) and \(dx\). |
| 6. | Given \(y = 1/\sqrt{x}\), express \(\Delta y\) and \(dy\) as functions of \(\Delta x\) and \(dx\). |
| 7. | Find \(d(x^{2} + 1/x^{2})\). |
| 8. | Find \(d(x - 1/x)\). |
| 9. | Find the equation of the line tangent to the curve \(y = 1/(x - 2)\) at the point \((1, -1)\). |
| 10. | Find the equation of the line tangent to the curve \(y = 1 + x \sqrt{x}\) at the point \((1, 2)\). |
| 11. | Find \(dy/dx\) where \(y = -3x^{3} - 5x + 2\). |
| 12. | Find \(dy/dx\) where \(y = (2x - 5)^{-2}\). |
| 13. | Find \(ds/dt\) where \(s = (3t + 4)(t^{2} - 5)\) |
| 14. | Find \(ds/dt\) where \(s = (4t^{2} - 6)^{-1} + (1 - 2t)^{-2}\). |
| 15. | Find \(du/dv\) where \(u = (2v^{2} - 5v + 1)/(v^{3} - 4)\). |
| 16. | Find \(du/dv\) where \(u = (v + (1/v))/(v - (1/v))\). |
| 17. | Find \(dy/dx\) where \(y = x^{1/2} + 4x^{3/2}\). |
| 18. | Find \(dy/dx\) where \(y = (1 + \sqrt{x})^{2}\). |
| 19. | Find \(dy/dx\) where \(y = x^{1/3} - x^{-1/4}\). |
| 20. | Find \(dy/dx\) where \(y = e^{x} \cos^{2} x\). |
| 21. | Find \(dy/dx\) where \(x = \sqrt{y} + y^{2}, \ y > 0\). |
| 22. | Find \(dy/dx\) where \(x = y^{-1/2} + y^{-1}, \ y > 0\). |
| 23. | Find \(dy/dx\) where \(y = \sqrt{1 - 3x}\). |
| 24. | Find \(dy/dx\) where \(y = \sin(2 + \sqrt{x})\). |
| 25. | Find \(dy/dx\) where \(y = u^{-1/2}, \ u = 5x + 4\). |
| 26. | Find \(dy/dx\) where \(y = u^{5}, \ u = 2 - x^{3}\). |
| 27. | Find the slope \(dy/dx\) of the path of a particle moving so that \(y = 3t + \sqrt{t}, \ x = (1/t) - t^{2}\). |
| 28. | Find the slope \(dy/dx\) of the path of a particle moving so that \(y = \sqrt{4t - 5}, \ x = \sqrt{3t + 6}\). |
| 29. | Find \(d^{2}y/dx^{2}\) where \(y = \sqrt{4x - 1}\). |
| 30. | Find \(d^{2}y/dx^{2}\) where \(y = x/(x^{2} + 2)\). |
| 31. | An object moves so that \(s = t \sqrt{t + 3}\). Find the velocity \(v = ds/dt\) and the acceleration \(a = d^{2}s/dt^{2}\). |
| 32. | Find \(dy/dx\) by implicit differentiation when \(x + y + 2x^{2} + 3y^{3} = 2\). |
| 33. | Find \(dy/dx\) by implicit differentiation when \(3xy^{3} + 2x^{3}y = 1\). |
| 34. | Find the slope of the line tangent to the curve \(2x \sqrt{y} - y^{2} = \sqrt{x}\) at \((1, 1)\). |
| \(\square\) 35. | Find the derivative of \(f(x) = \left|x^{2} - 1\right|\). |
| \(\square\) 36. | Find the derivative of the function \[f(x) = \begin{cases} 1 &\quad \text{if } x \text{ is an integer}, \\ 0 &\quad \text{otherwise}. \end{cases} \nonumber\] |
| \(\square\) 37. | Let \(f(x) = (x- c)^{4/3}\). Show that \(f'(x)\) exists for all real \(x\) but that \(f''(c)\) does not exist. |
| \(\square\) 38. | Let \(n\) be a positive integer and \(c\) a real number. Show that there is a function \(g(x)\) which has an \(n\)th derivative at \(x = c\) but does not have an \((n + 1)\)st derivative at \(x = c\). That is, \(g^{(n)}(c)\) exists but \(g^{(n+ 1)}(c)\) does not. |
| \(\square\) 39. | (a) Let \(u = |x|, \ y = u^{2}\). Show that at \(x = 0\), \(dy/dx\) exists even though \(du/dx\) does not. (b) Let \(u = x^{4}, \ y = |u|\). Show that at \(x = 0\), \(dy/dx\) exists even though \(dy/du\) does not. |
| \(\square\) 40. | Suppose \(g(x)\) is differentiable at \(x = c\) and \(f(x) = |g(x)|\). Show that (a) \(f'(c) = g'(c)\) if \(g(c) > 0\), (b) \(f'(c) = -g'(c)\) if \(g(c) < 0\), (c) \(f'(c) = 0\) if \(g(c) = 0\) and \(g'(c) = 0\), (d) \(f'(c)\) does not exist if \(g(c) = 0\) and \(g'(c) \neq 0\). |
| \(\square\) 41. | Prove by induction that for every positive integer \(n\), \(n < 2^{n}\). |
| \(\square\) 42. | Prove by induction that the sum of the first \(n\) odd positive integers is equal to \(n^{2}\), \[1 + 3 + 5 + \cdots + (2n-1) = n^{2}. \nonumber\] |