Question

Find the linear approximation for f(x) = 12x3 + 3x2 + x + 2 at x= 1.

Answer 1

To find the linear approximation for the function f(x) = 12x^3 + 3x^2 + x + 2 at x = 1, we can use the equation for the linear approximation. First, we need to find the derivative of the function and then plug in the values into the formula. The linear approximation is f(1).

Step-by-step explanation:

To find the linear approximation for the function f(x) = 12x^3 + 3x^2 + x + 2 at x = 1, we can use the equation for the linear approximation:

() ≈ () + '()( − )

First, we need to find '(), which is the derivative of the function. Taking the derivative of f(x) gives us:

'() = 36^2 + 6 + 1

Next, we plug in x = 1 into the first equation:

(1) ≈ (1) + '(1)(1 − 1)

Simplifying, we have:

(1) ≈ (1)

So, the linear approximation for f(x) at x = 1 is f(1).

Answer 2

y = 43x − 25

Step-by-step explanation:

Evaluate the function at x=1:

f(x) = 12x³ + 3x² + x + 2

f(1) = 12 + 3 + 1 + 2

f(1) = 18

Find the slope of the tangent line at x=1:

f'(x) = 36x² + 6x + 1

f'(1) = 36 + 6 + 1

f'(1) = 43

Point-slope form:

y − y₀ = m (x − x₀)

y − 18 = 43 (x − 1)

Convert to slope-intercept form:

y − 18 = 43x − 43

y = 43x − 25

Graph:

desmos.com/calculator/giumpkkphr