Chapter 2: Q4E (page 77)
4: From the given graph of \(g\), state the numbers at which \(g\)
is discontinuous and explain why.
Points of discontinuity are\(x = - 2, - 1,0,1\)
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Use equation 5 to find \(f'\left( a \right)\) at the given number \(a\).
\(f\left( x \right) = \frac{{\bf{1}}}{{\sqrt {{\bf{2}}x + {\bf{2}}} }}\), \(a = {\bf{1}}\)
36: Prove that \(\mathop {\lim }\limits_{x \to 2} \frac{1}{x} = \frac{1}{2}\).
(a) The curve with the equation \({y^2} = 5{x^4} - {x^2}\)is called akampyle of Eudoxus. Find and equation of the tangent line to this curve at the point\(\left( {1,2} \right)\)
(b) Illustrate part\(\left( a \right)\)by graphing the curve and the tangent line on a common screen. (If your graph device will graph implicitly defined curves, then use that capability. If not, you can still graph this curve by graphing its upper and lower halves separately.)
The point \(P\left( {{\bf{1}},{\bf{0}}} \right)\) lies on the curve \(y = {\bf{sin}}\left( {\frac{{{\bf{10}}\pi }}{x}} \right)\).
a. If Qis the point \(\left( {x,{\bf{sin}}\left( {\frac{{{\bf{10}}\pi }}{x}} \right)} \right)\), find the slope of the secant line PQ (correct to four decimal places) for \(x = {\bf{2}}\), 1.5, 1.4, 1.3, 1.2, 1.1, 0.5, 0.6, 0.7, 0.8, and 0.9. Do the slopes appear to be approaching a limit?
b. Use a graph of the curve to explain why the slopes of the secant lines in part (a) are not close to the slope of the tangent line at P.
c. By choosing appropriate secant lines, estimate the slope of the tangent line at P.
19-32: Prove the statement using the \(\varepsilon ,\delta \) definition of a limit.
30. \(\mathop {\lim }\limits_{x \to 2} \left( {{x^2} + 2x - 7} \right) = 1\)
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