Chapter 2: Q. 68 (page 136)
Calculate each of the limits
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
The gravitational force exerted by the planet Earth on a unit mass at a distance r from the center of the planet is
\(F\left( r \right) = \left\{ {\begin{array}{*{20}{c}}{\frac{{GMr}}{{{R^{\bf{3}}}}}}&{{\bf{if}}\,\,\,r < R}\\{\frac{{GM}}{{{r^{\bf{2}}}}}}&{{\bf{if}}\,\,\,r \ge R}\end{array}} \right.\)
Where M is the mass of Earth, R is its radius, and G is the gravitational constant. Is F a continuous function of r?
A roast turkey is taken from an oven when its temperature has reached \({\bf{185}}\;^\circ {\bf{F}}\) and is placed on a table in a room where the temperature \({\bf{75}}\;^\circ {\bf{F}}\). The graph shows how the temperature of the turkey decreases and eventually approaches room temperature. By measuring the slope of the tangent, estimate the rate of change of the temperature after an hour.
Which of the following functions \(f\) has a removable discontinuity at \(a\)? If the discontinuity is removable, find a function \(g\) that agrees with \(f\) for \(x \ne a\)and is continuous at \(a\).
(a) \(f\left( x \right) = \frac{{{x^4} - 1}}{{x - 1}},a = 1\)
(b) \(f\left( x \right) = \frac{{{x^3} - {x^2} - 2x}}{{x - 2}},a = 2\)
(c)\(f\left( x \right) = \left [{\sin x} \right],a = \pi \)
Each limit represents the derivative of some function f at some number a. State such as an f and a in each case.
\(\mathop {{\bf{lim}}}\limits_{h \to {\bf{0}}} \frac{{{\bf{tan}}\left( {\frac{\pi }{{\bf{4}}} + h} \right) - {\bf{1}}}}{h}\)
Sketch the graph of the function f for which \(f\left( {\bf{0}} \right) = {\bf{0}}\), \(f'\left( {\bf{0}} \right) = {\bf{3}}\), \(f'\left( {\bf{1}} \right) = {\bf{0}}\), and \(f'\left( {\bf{2}} \right) = - {\bf{1}}\).
What do you think about this solution?
We value your feedback to improve our textbook solutions.
