Question

A camera/radar clocks a sports car at 65 mph. Eight minutes later, another camera/radar, 10 miles away, clocks the same car at 60 mph. Showing your work, how can you prove that this sports car exceeded the 65 mph speed limit at some time during the eight minutes. (Assume the position function is differentiable)

Answer 1

average velocity = total distance/total time = 10miles/8minutes = 75mph

Answer 2

distance traveled = average velocity * time = integral(v(t) dt)
=75*8=integral(v(t)dt from t=0 to t=8)
using the mean value thm or something similar, v(t)=75 for some t between 0 and 8

Answer 3

\[\int\limits_{a}^{b}f = F(b)-F(a)\] where F is the antiderivative of f; antiderivative is the same as indefinite integral

Answer 4

have you learned about definite & indefinite integrals yet

Answer 5

have you learned the mean value theorem?

Answer 6

actually you don't need integrals for this, just use the mean value theorem directly on x(t):
x'(c)=(x(b)-x(a))/(b-a) for some time c

Answer 7

let a=0 and x(0)=0
then b=8 and x(8)=10
remember, x is the position
x' is the velocity

Answer 8

yup :)

Answer 9

you have x'(c)=(x(8)-x(0))/(8-0)=(10miles-0miles)/(8minutes-0minutes)=1.25miles/minutes=75mph for some 0<c<8

Answer 10

looks good, remember to convert from miles/minute to miles/hour though

Answer 11

cool :)