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$f(x) = \sqrt{x} - 1$ If the index is even, determine at least $3$ points, including the key point. If the index is odd, determine at least $5$ points.

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Find values of derivatives using limits

Full solution

Q.

f(x) = \sqrt{x} - 1

If the index is even, determine at least

3

points, including the key point. If the index is odd, determine at least

5

points.

Identify Nature of Index: Identify the nature of the index in the function $f(x) = \sqrt{x} - 1$ . The index of the square root is $2$ , which is even.
Choose Perfect Squares: Choose $x$ -values that are perfect squares and greater than or equal to $0$ , since the square root of a negative number is not real. $\newline$ Let's choose $x = 0$ , $x = 1$ , and $x = 4$ as our $x$ -values to find the corresponding $y$ -values.
Calculate $y$ for $x=0$ : Calculate the $y$ -value for $x = 0$ . $f(0) = \sqrt{0} − 1 = 0 − 1 = -1$ The point is $(0, -1)$ .
Calculate $y$ for $x=1$ : Calculate the $y$ -value for $x = 1$ . $f(1) = \sqrt{1} − 1 = 1 − 1 = 0$ The point is $(1, 0)$ .
Calculate $y$ for $x=4$ : Calculate the $y$ -value for $x = 4$ . $f(4) = \sqrt{4} − 1 = 2 − 1 = 1$ The point is $(4, 1)$ .

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