‘Use differential form to approximate %65 (cube root)-_’
The Correct Answer and Explanation is:
To approximate the cube root of 65 using the differential form, we start by defining the function f(x)=x3f(x) = \sqrt[3]{x}f(x)=3x, which is the cube root of xxx.
Step 1: Choose a point of approximation
Since we are approximating 653\sqrt[3]{65}365, we choose a point close to 65 where the cube root is easy to calculate. A natural choice is x=64x = 64x=64, since 643=4\sqrt[3]{64} = 4364=4.
Step 2: Compute the derivative of f(x)f(x)f(x)
We need the derivative of f(x)=x1/3f(x) = x^{1/3}f(x)=x1/3. Using the power rule, the derivative is:f′(x)=13x−2/3f'(x) = \frac{1}{3} x^{-2/3}f′(x)=31x−2/3
Step 3: Apply the differential formula
The differential approximation formula is:f(x+Δx)≈f(x)+f′(x)Δxf(x + \Delta x) \approx f(x) + f'(x) \Delta xf(x+Δx)≈f(x)+f′(x)Δx
Where:
- f(x)=x3f(x) = \sqrt[3]{x}f(x)=3x is the original function.
- f′(x)=13x−2/3f'(x) = \frac{1}{3} x^{-2/3}f′(x)=31x−2/3 is the derivative.
- Δx=65−64=1\Delta x = 65 – 64 = 1Δx=65−64=1 is the change in xxx.
Step 4: Calculate the approximation
At x=64x = 64x=64, we know that f(64)=4f(64) = 4f(64)=4. Now, we evaluate f′(64)f'(64)f′(64):f′(64)=13×64−2/3f'(64) = \frac{1}{3} \times 64^{-2/3}f′(64)=31×64−2/3
Since 641/3=464^{1/3} = 4641/3=4, 64−2/3=142=11664^{-2/3} = \frac{1}{4^2} = \frac{1}{16}64−2/3=421=161. Thus:f′(64)=13×116=148f'(64) = \frac{1}{3} \times \frac{1}{16} = \frac{1}{48}f′(64)=31×161=481
Now, we can approximate f(65)f(65)f(65) as:f(65)≈f(64)+f′(64)×Δx=4+148×1=4+148f(65) \approx f(64) + f'(64) \times \Delta x = 4 + \frac{1}{48} \times 1 = 4 + \frac{1}{48}f(65)≈f(64)+f′(64)×Δx=4+481×1=4+481
So:f(65)≈4+0.0208=4.0208f(65) \approx 4 + 0.0208 = 4.0208f(65)≈4+0.0208=4.0208
Conclusion:
The approximation for 653\sqrt[3]{65}365 using the differential form is 4.02084.02084.0208.
Explanation:
In this method, the differential approximation is based on linearizing the function at a point where the value is easy to calculate. By using the derivative to estimate how much the function changes as xxx increases from 64 to 65, we can get a very close approximation for 653\sqrt[3]{65}365, which in this case is approximately 4.0208.
