to the curve y=f(x) at the point having given value of x.

1. f(x)=(2x-3)^(5/2), x=2

2. f(x)=xsqrt(25-x^2), x=4

3. f(x) = (sqrt(5+2x))/x, x=2

Can I please have an explanation to what a normal line is and how to solve these step by step? Thanks

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The normal line is a line that is perpendicular to the tangent line and passes through the same point on f(x) as the tangent line does.

I will solve the first one for you. The second and third problems are similar!

1)

Taking derivatives gives:

f'(x) = (2)(5/2)(2x – 3)^(5/2 – 1)

==> f'(x) = 5(2x – 3)^(3/2).

At x = 2, we see that the slope of the tangent line is:

f'(2) = 2[2(2) – 3]^(5/2) = 2(1)^(5/2) = 2.

(This gives the slope of the normal line to be -1/2)

Since f(x) passes through (2, 1), we see that the tangent line is:

y – 1 = 2(x – 2)

==> y = 2x – 3.

Then, the normal line is:

y – 1 = (-1/2)(x – 2)

==> y = (-1/2)x + 2.

I hope this helps!

Anonymous