12 th grade AP Calc Class-student

Let f be the function defined for all x > -5 and having the following properties. Find an expression for f(x).

i) f ''(x) = 1/ (x+5)1/3 for all x in the domain of f

ii) the line tangent to the graph of f at (4,2) has an angle of inclination of 45 degress.


First of all the second property contains two pieces of information. The fact that (4,2) is on the graph of the function implies that f(4) = 2. Second a line with angle of inclination 45o has a slope of 1. Hence f '(4) = 1.

The first property says that f ''(x) = (x+5)-1/3. Thus by antidifferentiation,

f '(x) =(3/2) (x+5)2/3 + C,
where C is a constant.

Use the fact that f '(4) = 1 to find C.

Antidifferentiate again to find an expression for f(x) and then use the fact that f(4) = 2 to evaluate the constand that arises in the andoidifferentiation.

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