Question

Solve using Taylor Polynomials and derivatives ONLY! Two bicyclists are racing down a straight road. At one point in time the bicyclists are right next to each other. At this moment, Bicyclist A has a velocity of 16 miles/hr and an acceleration of 32 miles/hr^2, and Bicyclist B has a velocity of 19 miles/hr and an acceleration of 5 miles/hr^2. Which bicyclist do you expect to be in front 15 minutes (1/4 hour) later? Explain your reasoning.

Answer 1

Let f(t) denote the distance of the cyclists. At t=0,

$f_A(0)=0, f_A'(0)=16, f_A''(0)=32\newline f_B(0)=0, f_B'(0)=19, f_B''(0)=5$

where denotes the velocity at $t=0$ and denotes the acceleration at $t=0$ .

According to Taylor's polynomial, we have

$f_A(t)=f_A(0)+tf_A'(0)+\frac{t^2}{2}f_A''(0)\newline f_B(t)=f_B(0)+tf_B'(0)+\frac{t^2}{2}f_B''(0)$

Substituting the values and calculating at $t=15$ would give us the distance travelled by cyclists. After calculation, we observe that the cyclist A is farther ahead than B. (Please calculate)