Consider the function f(t) = 7t^2 + 14t, measured in seconds. a. Calculate the average velocity of the function during the following intervals: i. from t = 1 to t = 2 ii. from t =... Consider the function f(t) = 7t^2 + 14t, measured in seconds. a. Calculate the average velocity of the function during the following intervals: i. from t = 1 to t = 2 ii. from t = 1 to t = 1.01 iii. from t = 1 to t = 1.1 b. Use the results of part a to approximate the instantaneous velocity when t = 1. Read more

Steps to Solve

1. Calculate the function values at the endpoints To find the average velocity over an interval, we need to evaluate the function at the endpoints.

• For interval (i), from $t = 1$ to $t = 2$:

  • $f(1) = 7(1^2) + 14(1) = 7 + 14 = 21$
  • $f(2) = 7(2^2) + 14(2) = 7(4) + 28 = 28 + 28 = 56$

• For interval (ii), from $t = 1$ to $t = 1.01$:

  • $f(1) = 21$ (from above)
  • $f(1.01) = 7(1.01^2) + 14(1.01) = 7(1.0201) + 14.14 \approx 7.1407 + 14.14 \approx 21.2807$

• For interval (iii), from $t = 1$ to $t = 1.1$:

  • $f(1) = 21$ (from above)
  • $f(1.1) = 7(1.1^2) + 14(1.1) = 7(1.21) + 15.4 = 8.47 + 15.4 \approx 23.87$

2. Calculate the average velocities for each interval The average velocity is calculated using the formula: $$\text{Average velocity} = \frac{f(b) - f(a)}{b - a}$$

• For interval (i): $$\text{Average velocity} = \frac{56 - 21}{2 - 1} = \frac{35}{1} = 35$$

• For interval (ii): $$\text{Average velocity} = \frac{21.2807 - 21}{1.01 - 1} = \frac{0.2807}{0.01} \approx 28.07$$

• For interval (iii): $$\text{Average velocity} = \frac{23.87 - 21}{1.1 - 1} = \frac{2.87}{0.1} \approx 28.7$$

3. Approximate the instantaneous velocity at $t = 1$ The instantaneous velocity can be estimated by taking the limit of the average velocity as the interval approaches zero. We will use the results from (ii) and (iii):

Using the average velocities from (ii) and (iii): $$\text{Instantaneous velocity} \approx \text{average velocity as interval shrinks} \approx 28.07 \text{ or } 28.7$$

By extending this approximation further, we could assume the instantaneous velocity approaches around 28.5.