+280 
I need to be able to know how to do (3.) for an exam next week. I just dont know if i put that function into the one above or do i just solve it. so i would get 0 for the slope.
+9479 g(x) = \(5-\sqrt{4-x}\)
When x = 3 , the slope of the curve \(=\,\lim\limits_{h\to0}\frac{g(3+h)-g(3)}{h} \\~\\ =\,\lim\limits_{h\to0}\frac{(5-\sqrt{4-(3+h)})-(5-\sqrt{4-3})}{h} \\~\\ =\,\lim\limits_{h\to0}\frac{5-\sqrt{4-3-h}-4}{h} \\~\\ =\,\lim\limits_{h\to0}\frac{1-\sqrt{1-h}}{h} \\~\\ =\,\lim\limits_{h\to0}(\,\frac{1-\sqrt{1-h}}{h}\,)\,(\,\frac{1+\sqrt{1-h}}{1+\sqrt{1-h}}\,) \\~\\ =\,\lim\limits_{h\to0}\frac{1-(1-h)}{h+h\sqrt{1-h}} \\~\\ =\,\lim\limits_{h\to0}\frac{h}{h+h\sqrt{1-h}} \\~\\ =\,\lim\limits_{h\to0}\frac{1}{1+\sqrt{1-h}} \\~\\ =\,\frac{1}{1+\sqrt{1-0}} \\~\\ =\,\frac12\)
Look at the graph and see that it does look like that line has a slope of \(\frac12\) .
We want an equation of a line that has a slope of \(\frac12\) and passes through the point ( 3, g(3) ) .
g(3) = \(5-\sqrt{4-3}\) = 5 - 1 = 4
So the equation of our line in point - slope form is
y - 4 = \(\frac12\)(x - 3)
And in slope - intercept form, this is... y = \(\frac12\)x + \(\frac52\)
+9479 g(x) = \(5-\sqrt{4-x}\)
When x = 3 , the slope of the curve \(=\,\lim\limits_{h\to0}\frac{g(3+h)-g(3)}{h} \\~\\ =\,\lim\limits_{h\to0}\frac{(5-\sqrt{4-(3+h)})-(5-\sqrt{4-3})}{h} \\~\\ =\,\lim\limits_{h\to0}\frac{5-\sqrt{4-3-h}-4}{h} \\~\\ =\,\lim\limits_{h\to0}\frac{1-\sqrt{1-h}}{h} \\~\\ =\,\lim\limits_{h\to0}(\,\frac{1-\sqrt{1-h}}{h}\,)\,(\,\frac{1+\sqrt{1-h}}{1+\sqrt{1-h}}\,) \\~\\ =\,\lim\limits_{h\to0}\frac{1-(1-h)}{h+h\sqrt{1-h}} \\~\\ =\,\lim\limits_{h\to0}\frac{h}{h+h\sqrt{1-h}} \\~\\ =\,\lim\limits_{h\to0}\frac{1}{1+\sqrt{1-h}} \\~\\ =\,\frac{1}{1+\sqrt{1-0}} \\~\\ =\,\frac12\)
Look at the graph and see that it does look like that line has a slope of \(\frac12\) .
We want an equation of a line that has a slope of \(\frac12\) and passes through the point ( 3, g(3) ) .
g(3) = \(5-\sqrt{4-3}\) = 5 - 1 = 4
So the equation of our line in point - slope form is
y - 4 = \(\frac12\)(x - 3)
And in slope - intercept form, this is... y = \(\frac12\)x + \(\frac52\)
