Find dy for the given values of x and Δx.
y=4/x^4+3√x; x=4, Δx=.5
dy/dx = -16 x^-5 + 1.5 x^-.5
so
dy/dx here = -.0156 + .75 = .7344
so
dy = .7344* .5 = .367
y = 4 x^-4 + 3 x^.5
dy/dx = (-20) x^-5 + 1.5 x^-.5
delta y=(-20/x^5)delta x+(1.5/x^.5)delta x
if x = 4 then x^5 = 1024 and x^.5 = 2
so
delta y =(-20/1024 + 1.5/2 ) .5
= .365
=========================
now exact
y1 = .003906 + 6 = 6.003906
y2 = .009755 + 6.3640
y2-y1 = delta y = .360
not bad
Where did you made error on the first one? did you typed wrong numbers or problem.
if y = x^-4
dy/dx = -4 x^-5
not
-5 x^-5
To find dy for the given values of x and Δx, we can use the differential calculus concept of finding the derivative.
First, let's find the derivative of the given function y with respect to x:
y = 4/x^4 + 3√x
Using the power rule and the chain rule, we can differentiate each term separately.
For the first term, 4/x^4, we can rewrite it as 4x^(-4) to apply the power rule:
d/dx (4x^(-4)) = -4 * 4x^(-5) = -16x^(-5) = -16/x^5
For the second term, 3√x, we can rewrite it as 3x^(1/2) to apply the power rule:
d/dx (3x^(1/2)) = 1/2 * 3x^(-1/2) = 3/2x^(1/2) = 3/(2√x)
Now, let's calculate dy by substituting the values of x and Δx into the derivative:
x = 4
dy = (-16/x^5) * Δx + (3/(2√x)) * Δx
Substituting x = 4 and Δx = 0.5:
dy = (-16/4^5) * 0.5 + (3/(2√4)) * 0.5
= (-16/1024) * 0.5 + (3/(2*2)) * 0.5
= (-1/64) + (3/4) * 0.5
= -0.015625 + 0.375
= 0.359375
Hence, dy for the given values of x = 4 and Δx = 0.5 is 0.359375.