Problem
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 \[ f^{\prime}(4)=\lim _{h \rightarrow 0} \frac{f(4+h)-f(4)}{h}= \] (c) Use your answer from part (b) to find the equation of the tangent line to the curve at the point $(4, f(4))$. \[ y= \]
Solution
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 ：\(f^\prime(4) = \lim_{h \rightarrow 0} \frac{f(4+h)-f(4)}{h} = \frac{3}{2\sqrt{4}}\)
Step 5 ：\(y - 2 = \frac{3}{2\sqrt{4}}(x - 4)\)
Step 6 ：\(\boxed{y - 2 = \frac{3}{4}(x - 4)}\)
