using the alternative form of the derivative in exercises 69 - 76, use the alternative form of the…
using the alternative form of the derivative in exercises 69 - 76, use the alternative form of the derivative to find the derivative at x = c, if it exists.\n69. f(x)=x³ + 2x² + 1, c = - 2\n70. g(x)=x² - x, c = 1\n71. g(x)=√|x|, c = 0\n72. f(x)=3/x, c = 4\n73. f(x)=(x - 6)^(2/3), c = 6\n74. g(x)=(x + 3)^(1/3), c = - 3\n75. h(x)=|x + 7|, c = - 7\n76. f(x)=|x - 6|, c = 6
Answer
Explanation:
Step1: Recall the alternative - form of the derivative
The alternative form of the derivative is $f^\prime(c)=\lim_{x\rightarrow c}\frac{f(x)-f(c)}{x - c}$.
Step2: For $f(x)=x^{3}+2x^{2}+1$ and $c = - 2$
First, find $f(c)$: $f(-2)=(-2)^{3}+2(-2)^{2}+1=-8 + 8+1 = 1$. Then, find $\frac{f(x)-f(c)}{x - c}$: $\frac{(x^{3}+2x^{2}+1)-1}{x+2}=\frac{x^{3}+2x^{2}}{x + 2}=\frac{x^{2}(x + 2)}{x + 2}=x^{2}$ for $x\neq - 2$.
Step3: Find the limit
$\lim_{x\rightarrow - 2}\frac{f(x)-f(-2)}{x+2}=\lim_{x\rightarrow - 2}x^{2}=(-2)^{2}=4$.
Answer:
4