Question

I’m actually confused about this

Answer 1

(a) L(x) = 5x + 1

(b) L(1.1) = 6.5

(c) error ≈ 0.00753

Explanation:

To address the question, we will follow a systematic approach to find the equation of the tangent line to the function at x = 1, use the tangent line to approximate the value of the function at x = 1.1, and then compute the actual value of the function at x = 1.1 to determine the error in the linear approximation.

Here is our given function:

`f(x) = \sqrt[3]{180x^3+36}`

Note: At x = 1, f(1) = 6

`\hrulefill`

Firstly, we calculate the derivative of the given function, which represents the slope of the tangent line at any point 'x'.

`\Longrightarrow (d)/(dx)\left[f(x) = \sqrt[3]{180x^3+36}\right]\\\\\\\\\Longrightarrow (d)/(dx)[f(x)] = (d)/(dx)\left[\sqrt[3]{180x^3+36}\right]\\\\\\\\\Longrightarrow f'(x)= (d)/(dx)\left[(180x^3+36)^{(1)/(3)}\right]\\\\\\\\\Longrightarrow f'(x)= (1)/(3) (180x^3+36)^{(1)/(3)-1} \cdot (d)/(dx)[180x^3+36]\\\\\\\\\Longrightarrow f'(x)= (1)/(3) (180x^3+36)^{-(2)/(3)} \cdot(540x^2)\\\\\\\\`

`\therefore f'(x)= \frac{180x^2}{(180x^3+36)^{(2)/(3)}}`

Plugging in x = 1 to find the slope:

`\Longrightarrow f'(1)= \frac{180(1)^2}{(180(1)^3+36)^{(2)/(3)}}\\\\\\\\\therefore f'(1)=5`

With this slope, we find the equation of the tangent line at x = 1 by using the point-slope form, y - y₁ = m(x - x₁).

We have,

• m = 5
• (x₁, y₁) = (1, 6)

Plugging our values in:

`\Longrightarrow y - 6 = 5(x - 1)\\\\\\\\\Longrightarrow y - 6 = 5x - 5\\\\\\\\\therefore \boxed{L(x) = 5x + 1}`

Thus, our tangent line at x = 1 is L(x) = 5x + 1. Using our equation of the tangent line, plug in x = 1.1 to approximate the value of 'y':

`\Longrightarrow L(1.1) = 5(1.1) + 1\\\\\\\\\Longrightarrow L(1.1) = 5.5 + 1\\\\\\\\\therefore \boxed{L(1.1)=6.5}`

To determine the actual value of 'y' plug in x = 1.1 into f(x) and approximate using a calculator:

`\Longrightarrow f(1.1) = \sqrt[3]{180(1.1)^3+36}\\\\\\\\\therefore f(1.1) \approx 6.50753 \text{ (rounded to 5 d.p.)}`

We can calculate the error as follows:

`\Longrightarrow \big| \big|\text{error} \big|\big| = \big|6.5 - 6.50753 \big|\\\\\\\\\therefore \boxed{\text{error}\approx 0.00753}`

Thus, the error is approximately 0.00753.