Question

Let f(x) = (x − 1)², g(x) = e^−2x, and h(x) = 1 + ln(1 − 2x).

(a) Find the linearizations of f, g, and h at 

a = 0.

L_f (x) = 

 

L_g(x) = 

 

L_h(x) = 

(b) Graph f, g, and h and their linear approximations. For which function is the linear approximation best? For which is it worst? Explain.

The linear approximation appears to be the best for the function  ? f g h since it is closer to  ? f g h for a larger domain than it is to  - Select - f and g g and h f and h . The approximation looks worst for  ? f g h since  ? f g h moves away from L faster than  - Select - f and g g and h f and h do.

Use differentials to estimate the amount of paint needed to apply a coat of paint 0.04 cm thick to a hemispherical dome with diameter 44 m. (Round your answer to two decimal places.)
  m³  

Answer 1

let y = f(x) is the function to find local linear approximation at x= x₀ we need to find equation of tangent to the given curve

at x = x₀

equation of tangent line at x= x₀

y - y₀ = f '(x) ( x - x₀)

when x goes toward x₀ , y from the line goes towards y from the curve

so when x is almost equal to x₀ , y from the line almost equal to y from the curve

so we can conclude at x= x₀ approximate value of y [y= f(x) ] from the curve is equal to value of y from the line

i.e. y = y₀ + f'(x₀) ( x- x₀ )

so local linear approximation of the curve y = f(x) at x= x₀ can be written as

L(x) = f(x₀) + f'(x₀) ( x- x₀ ) ........(1)