Linear Approximation Help!

Question

Let g(x) = Integal of -25 to x^2 f(t) dt
Use the techniques of linear approximation to approximate g(4.8) given the following information about f:
f is continuous on (-00,00)
f is an odd function
f(25) =1/8

Answer 1

g(x) = ∫f(t)dt from -25 to x²

 

So, g(5) = ∫f(t)dt from -25 to 25

 

              = 0 (because f(t) is an odd function)

 

By the Fundamental Theorem of Calculus, g'(x) = f(x²)(2x)

 

                                                         So, g'(5) = f(25)(10)

 

                                                                       = (1/8)(10) = 5/4

 

g(x) ≈ L(x) = g(a) + g'(a)(x-a)

 

g(4.8) ≈ L(4.8) = g(5) + g'(5)(4.8-5)

 

                        = 0 + (5/4)(-1/5) = -1/4