Consider the function $f (x) = 3 \sqrt{x} - 4$ .
(a) Simplify the following difference quotient as much as possible
$\frac{f (4 + h) - f (4)}{h}$
(b) Use your result from (a) and the limit definition of the derivative to calculate
$f^{'} (4) = lim_{h \to 0} \frac{f (4 + h) - f (4)}{h} =$
(c) Use your answer from part (b) to find the equation of the tangent line to the curve at the point $(4, f (4))$ .
$y =$

Answer

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Answer

$y - 2 = \frac{3}{4} (x - 4)$

Steps

Step 1 ： $f (4 + h) = 3 \sqrt{4 + h} - 4$

Step 2 ： $f (4 + h) - f (4) = 3 \sqrt{4 + h} - 4 - (3 \sqrt{4} - 4)$

Step 3 ： $\frac{f (4 + h) - f (4)}{h} = \frac{3 \sqrt{4 + h} - 4 - (3 \sqrt{4} - 4)}{h}$

Step 4 ： $f^{'} (4) = lim_{h \to 0} \frac{f (4 + h) - f (4)}{h} = \frac{3}{2 \sqrt{4}}$

Step 5 ： $y - 2 = \frac{3}{2 \sqrt{4}} (x - 4)$

Step 6 ： $y - 2 = \frac{3}{4} (x - 4)$