Respuesta :
Answer:
4 km/hr
Step-by-step explanation:
First, we can solve for the general form of the instantaneous velocity function by finding the derivative of the position function:
[tex]f'(t)=(t^3-2t^2+4)'[/tex]
↓ applying the sum/difference rule: [tex](f\pm g)' = f' \pm g'[/tex]
[tex]f'(t) = (t^3)' - (2t^2)' + (4)'[/tex]
↓ taking the constant out of the middle term
[tex]f'(t) = (t^3)' - 2(t^2)' + (4)'[/tex]
↓ applying the power rule: [tex](x^n)' = nx^{n-1}[/tex]
[tex]f'(t) = 3t^2 - 2(2t^1) + 0[/tex]
(the last term is zero because [tex]n=0[/tex], so [tex]nx^{n-1} = 0(x^{n-1}) = 0[/tex])
↓ simplifying the second term
[tex]f'(t) = 3t^2 - 4t[/tex]
↓ renaming f'(t) as v(t)
[tex]v(t) = 3t^2 - 4t[/tex]
Next, we can solve for the car's instantaneous velocity at t = 2 by plugging that t-value into the general form of the instantaneous velocity function that we just solved for:
[tex]v(2) = 3(2^2) - 4(2)[/tex]
[tex]v(2) = 12 - 8[/tex]
[tex]\boxed{v(2) = 4\text{ km}/\text{hr}}[/tex]
Final answer:
The instantaneous velocity of a car given the position function f(t) = t³ - 2t² + 4 at t = 2 hours is found by differentiating f(t) to get the velocity function v(t) and then solving for v(2), which is 4 kilometers per hour.
Explanation:
The instantaneous velocity of a car given the position function f(t) = t³ - 2t² + 4 at t = 2 hours is found by differentiating f(t) to get the velocity function v(t) and then solving for v(2), which is 4 kilometers per hour.
Instantaneous Velocity Calculation
The position function of a car moving along a mountain road is f(t) = t³ - 2t² + 4. To find the instantaneous velocity at t = 2 hours, we need to take the derivative of the position function with respect to time t, which gives us the velocity function. The velocity function v(t) is found using this derivative.
The derivative of f(t) is:
f'(t) = 3t² - 4t.
To find the velocity at t = 2 hours, we substitute 2 for t into the derivative function, which gives:
v(2) = 3(2)² - 4(2) = 12 - 8 = 4 kilometers per hour.
Therefore, the instantaneous velocity of the car at t = 2 hours is 4 kilometers per hour.
