017 10.0 points

Lance Armstrong is moving with speed
35 feet/sec and acceleration 4 feet/sec2 when he crosses the starting line of a stage in the Tour de France. Use a degree two Taylor polynomial T2(t) to estimate how far he travels in
the next second.
1. distance = 38 feet
2. distance = 37 feet
3. distance = 39 feet
4. distance = 41 feet
5. distance = 40 feet

Shouldn't it be 41?
My work:
f(t)=35t+2t^2
f'(t)=35+4t
f''(t)=4

39(x-1)/1! + 4(x-1)^2/2!
39+2=41 (41 is incorrect)

It's a Maclurian Series (a=0) so your work is just 35/1!+4/2!= 37.

To estimate how far Lance Armstrong travels in the next second, we can use a degree two Taylor polynomial.

Let's define the function f(t) as the distance traveled at time t. In this case, f(t) = 35t + 2t^2.

Using the Taylor polynomial formula for a second-degree polynomial (T2(t)), we need to evaluate the function and its derivatives at t = 1.

First, let's find f(1):
f(1) = 35(1) + 2(1)^2 = 35 + 2 = 37 feet.

Next, let's calculate the derivative f'(t):
f'(t) = 35 + 4t.

Now, let's evaluate f'(1):
f'(1) = 35 + 4(1) = 35 + 4 = 39.

Finally, let's find the second derivative f''(t):
f''(t) = 4.

Since we are using a second-degree polynomial, only the values of f(1), f'(1), and f''(1) are required.

Using the Taylor polynomial formula for a second-degree polynomial:
T2(t) = f(1) + f'(1)(t-1) + (1/2)f''(1)(t-1)^2

T2(t) = 37 + 39(t-1) + (1/2)(4)(t-1)^2
= 37 + 39(t-1) + 2(t-1)^2.

Now, let's calculate T2(t) at t = 2 (one second later):
T2(2) = 37 + 39(2-1) + 2(2-1)^2
= 37 + 39(1) + 2(1)^2
= 37 + 39 + 2
= 78.

Therefore, Lance Armstrong travels 78 feet in the next second.

None of the provided answer choices is 78, so none of them is correct.

To estimate how far Lance Armstrong will travel in the next second using a degree two Taylor polynomial, we can use the formula:

T2(t) = f(a) + f'(a) * (t - a) + f''(a) * (t - a)^2 / 2!

In this case, the given function is f(t) = 35t + 2t^2, where t represents time in seconds. The given values are:

Initial speed, f'(0) = 35 feet/sec
Acceleration, f''(0) = 4 feet/sec^2

To estimate the distance traveled by Lance Armstrong in the next second, we need to evaluate the polynomial T2(t) at t = 1.

First, let's get the values for f(a), f'(a), and f''(a). Since Armstrong crosses the starting line at time t = 0, we have:

f(0) = 35(0) + 2(0)^2 = 0 feet
f'(0) = 35 feet/sec
f''(0) = 4 feet/sec^2

Substituting these values into the Taylor polynomial formula, we have:

T2(t) = f(0) + f'(0) * (t - 0) + f''(0) * (t - 0)^2 / 2!

Simplifying, we get:

T2(t) = 0 + 35t + 2t^2 / 2!

Now, let's compute T2(1) to find the estimated distance traveled in the next second:

T2(1) = 0 + 35(1) + 2(1)^2 / 2!

T2(1) = 35 + 2 / 2

T2(1) = 35 + 1

T2(1) = 36 feet

Therefore, based on the given information and the calculation using the degree two Taylor polynomial, the estimated distance Lance Armstrong will travel in the next second is 36 feet. Thus, option 2 "distance = 37 feet" is the correct answer.