Question

Let f(x)=xsqrt(x+7) Answer the following questions.
Find the average slope of the function f on the interval [−7,0].
Verify the Mean Value Theorem by finding a number c in (−7,0) such that f′(c)=average slope

Answer 1

\[f(x)=x \sqrt{x+7}\]

Answer 2

muzzammil: are you familiar with the formula for the average slope of a function f(x)? If not, please look it up and write it down, as this formula is used repeatedly in defining the derivative of f(x).

Answer 3

You're interested in the values of f(x) at the interval endpoints x=-7 and x=0. Calculate these values and insert them into the formula for the average slope of a function. What next?

Answer 4

yeah its f(b)-f(a)/b-a

Answer 5

the average slope i got was sqrt14

Answer 6

That'll be correct if only you use additional parentheses: (f(b)-f(a))/(b-a). Sorry to be picky here, but not using those extra parentheses could cause you all kinds of trouble.

Answer 7

ok so it would be sqrt14

Answer 8

I'll be back with you in a moment. In the meantime, would you please find the derivative of f(x) = x*Sqrt(x+7).

Answer 9

yes its sqrt(x+7)+(1/2)(x+7)^(-1/2)(x)

Answer 10

I'm checking that out. Meanwhile, please go back to finding the average slope of f(x) over the given interval [-7,0]. f(-7) = ? f(0) = ? f(0)-f(-7) = ?

Answer 11

muzzammil: How are you doing?

Answer 12

i think both f(-7) and f(0) equal 0

Answer 13

That's right. So, what is the average slope of f(x) over the interval [-7,0]?

Answer 14

0

Answer 15

Right! Your derivative f'(x) is also correct. Substitute c for x in this derivative, set this derivative equal to 0 (from your previous result), and solve for c.

Answer 16

thanks it was -14/3

Answer 17

SO cool! So you have verified the Mean Value Theorem.

Answer 18

Here's what we've found: at x = -14/3 (which is within the interval [-7,0], the average slope of the function is equal to the slope of the tangent line to the graph.

Answer 19

All the best to you. Thanks for the medal!