Use Upper DeltaΔyalmost equals≈f prime left parenthesis x right parenthesisf′(x)Upper DeltaΔx to find a decimal approximation of the radical expression.
RootIndex 3 StartRoot 64.42 EndRoot364.42
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Part 1
What is the value found by using Upper DeltaΔyalmost equals≈f prime left parenthesis x right parenthesisf′(x)Upper DeltaΔ​x?

To approximate the value of \( \sqrt[3]{64.42} \) using the formula \( \Delta y \approx f'(x) \Delta x \), we first define the function:

\[
f(x) = \sqrt[3]{x}
\]

### Step 1: Calculate the derivative \( f'(x) \)

Using the power rule:

\[
f'(x) = \frac{1}{3} x^{-\frac{2}{3}} = \frac{1}{3 \sqrt[3]{x^2}}
\]

### Step 2: Choose a point near \( 64.42 \)

We'll use \( x = 64 \) because \( \sqrt[3]{64} = 4 \) is easy to compute.

### Step 3: Find \( f'(64) \)

Calculating \( f'(64) \):

\[
f'(64) = \frac{1}{3 \sqrt[3]{64^2}} = \frac{1}{3 \cdot 16} = \frac{1}{48}
\]

### Step 4: Determine \( \Delta x \)

Calculate \( \Delta x \):

\[
\Delta x = 64.42 - 64 = 0.42
\]

### Step 5: Calculate \( \Delta y \)

Now we can find \( \Delta y \):

\[
\Delta y \approx f'(64) \Delta x = \frac{1}{48} \cdot 0.42 \approx 0.00875
\]

### Step 6: Approximate \( f(64.42) \)

Now, we can approximate:

\[
f(64.42) \approx f(64) + \Delta y = 4 + 0.00875 \approx 4.00875
\]

### Final Result

Thus, the decimal approximation of \( \sqrt[3]{64.42} \) is approximately **4.00875**.

If you have any further questions or need more details, let me know! Here are some follow-up questions you might consider:

1. How would the approximation change with a different value for \( x \)?
2. What is the exact value of \( \sqrt[3]{64.42} \)?
3. How can you improve the accuracy of this approximation?
4. What are the applications of using linear approximation in calculus?
5. Can you explain the concept of differentials in this context?

**Tip:** Always choose a point \( x \) that is close to the value you want to approximate to improve accuracy.

Use Δy ≈ f′(x) Δx to find a decimal approximation of the radical expression. Find an approximation for the cube root of 64.42.

Learn how to approximate the cube root of 64.42 using the linear approximation formula Δy ≈ f′(x) Δx. This calculus problem involves differentiation and applying linear approximation to find a precise estimate for radical expressions. Suitable for high school calculus students in Grades 11-12.

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