Final answer:
To estimate the product of (1.02)^3(-3.02)^3, we use linear approximation for small changes around 1 and -3 and find the estimated value is -29.2324. We then compare this with the actual calculator value and use the formula to calculate the percentage error.
Step-by-step explanation:
To estimate the value of (1.02)^3(-3.02)^3 using a linear approximation, we can apply the formula for the derivative at a point, which gives us the best linear approximation near that point. For small changes in the input, we can assume the function behaves similarly to its tangent line. We will use the fact that for any function f(x) that is differentiable at x = a, we can say that f(x) ≈ f(a) + f'(a)(x-a).
The function we are considering here is f(x) = x^3. In this case, we're looking at points near 1 for the first part (1.02)^3 and -3 for the second part (-3.02)^3. The derivative of f(x) = x^3 is f'(x) = 3x^2. Therefore, for a small change around 1 and -3, we can use the approximations:
- (1.02)^3 ≈ 1^3 + 3(1^2)(1.02-1) = 1 + 3(0.02) = 1.06
- (-3.02)^3 ≈ (-3)^3 + 3(-3)^2(-3.02 + 3) = -27 + 3(9)(-0.02) = -27 - 0.54 = -27.54
Now, by multiplying these two approximations, we get an estimate:
(1.02)^3(-3.02)^3 ≈ 1.06 * (-27.54) ≈ -29.2324
To find the percentage error, we compare this with the value given by a calculator:
Actual value from calculator: (1.02)^3(-3.02)^3
Estimated value from linear approximation: -29.2324
Error = (Actual value - Estimated value) / Actual value * 100%