# Find an equation for the tangent line

suppose the function F is define by F(x)=[integral from 1 to √x] (2t-1)/(t+2)dt for all real numbers x≥0

A.) evaluate F(1)
B.) Evaluate F'(1)
C.)Find an equation for the tangent line to the graph of F at the point where x=1
D.) on what intervals is the function F increasing? justify your answer

suppose the function F is define by F(x)=[integral from 1 to √x] (2t-1)/(t+2)dt for all real numbers x≥0

A.) evaluate F(1)
B.) Evaluate F'(1)
C.)Find an equation for the tangent line to the graph of F at the point where x=1
D.) on what intervals is the function F increasing? justify your answer

You have

$$F(x) = \int_1^{\sqrt{x}} \frac{2t-1}{t+2}\ dt$$

(A) Write down exactly what F(1) is using your formula and stare at the limits of integration for a few seconds.
(B) Consider the Fundamental Theorem of Calculus (you will need to apply the chain rule, too)
(C) Should be easy once you have (A) and (B)
(D) How does the sign of the derivative of any function relate to whether the function is increasing/decreasing?

i know it should be simple, i dont know how to integrate 2t-1/t+2 help me out

Try substituting u=t+2.

ok what about part b i used the fundamental rule but what about finding it?

Mark44
Mentor

Try substituting u=t+2.
Alternatively, you can just divide 2t -1 by t + 2 using polynomial long division.

Mark44
Mentor

ok what about part b i used the fundamental rule but what about finding it?
Show us what you did in using the Fundamental Theorem of Calculus.

i know it should be simple, i dont know how to integrate 2t-1/t+2 help me out

Don't focus on what the integrand is; focus on the limits. Let's just call the integrand g(t). When x=1, you get

$$F(1) = \int_1^{1} g(t)\ dt$$

Again, stare at the limits of integration... can you visualize the "interval" over which you are integrating?

The point is that while you _can_ integrate, you actually do not need to do ANY sort of computation if you take a closer look at your integral.

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