Use the linear approximation (1+x)^k\approx 1+kx to find an approximation for the function f(x) for values of x near zero.
I need help for (3+3x)^(1/3). Please help me!
f'(x) = (3x+3)^(-2/3)
f'(0) = 3^(-2/3)
f(0) = 3^(1/3)
So, near x=0,
y-3^(1/3) = 3^(-2/3) x
you can see this graphically at wolframalpha.com if you enter
plot y = (3+3x)^(1/3),y = 3^(-2/3) x + 3^(1/3) for -1<x<1
Thanks a lot Steve!
To find an approximation for the function f(x) = (3+3x)^(1/3) using the linear approximation (1+x)^k ≈ 1 + kx, we can follow these steps:
Step 1: Identify the values needed for the linear approximation formula:
- Function to approximate: f(x) = (3+3x)^(1/3)
- Base value: a = 0 (since we are approximating values near zero)
- Small change or increment: Δx = x - a
- Exponent value: k = 1/3
Step 2: Apply the linear approximation formula:
- Replace (3+3x) with (1 + Δx) and 1/3 with k in the formula: (1+Δx)^k ≈ 1 + kΔx
- Rearrange the formula to match our function: (3+3x)^1/3 ≈ 1 + 1/3 * (x - 0)
Step 3: Simplify the linear approximation:
- Distribute the 1/3 to simplify the expression: (3+3x)^(1/3) ≈ 1 + x/3
Step 4: Interpret the approximation:
- The approximation for the function f(x) = (3+3x)^(1/3) is approximately 1 + x/3 for values of x near zero.
So, for values of x near zero, you can use the approximation 1 + x/3 to estimate the value of the function (3+3x)^(1/3).