Posted by **Adam Janky** on Thursday, March 10, 2011 at 9:53pm.

Consider the following.

y = (2x2 + 5)(x3 − 25x)

at (5, 0)

(a) At the indicated point, find the slope of the tangent line.

(b) Find the instantaneous rate of change of the function.

- calculus -
**MathMate**, Friday, March 11, 2011 at 8:07am
First check if (5,0) is on the line:

f(x)= (2x2 + 5)(x3 − 25x)

f(5)= (50+5)(125-125)=0 indeed.

a)Slope

Slope of the tangent line at (5,0) is the value of f'(x) at (5,0), namely f'(5).

So let's find f'(x).

We can find f'(x) as a product, or as an expanded polynomial. I choose the latter:

y = (2x2 + 5)(x3 − 25x)

=2x^5+5x^3-50x^3-125x^2

=2x^5-45x^3-125x^2

Differentiate term by term:

f'(x)=10x^4-135x2-250x

so

f'(5)=6250-3375-1250=1625

= slope of tangent at (5,0)

b.

instantaneous rate of change of the function is precisely f'(x)=dy/dx.

So the answer is the same as in a.

## Answer this Question

## Related Questions

- calculus - Consider the following. y = x2 − 9x x2 + 5x at (3, − 3 4...
- calculus - At the indicated point for the function, find the following. A ...
- calculus - i'm not sure how to do this. can someone help, please? thanks! ...
- calculus - To find the equation of a line, we need the slope of the line and a ...
- calculus - For the function y = 9x2 + 6x + 2, at the point x = 7, find the ...
- Calculus - Would you be able to help me with a couple of questions? thanks in ...
- Calculus - Use the four-step process to find the slope of the tangent line to ...
- Math - The point P(8, −3) lies on the curve y = 3/(7 − x). (a) If Q ...
- Pre-Calculus - Find the slope of the tangent line to the curve 2(x^2+y^2)^2=25(x...
- Calculus [finding slope of tangent line] - Find the slope of the tangent line to...