ages iol lac ploviel the correct order and rotated score of 0. You should always

Question

ages iol lac ploviel the correct order and rotated score of 0. You should always check questions are on separate pages. properly! Otherwise, it will receive a your submission using a computer. Be sure that separate 1. (10 points) Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed. Hint: Apply the Mean Value Theorem to f(t) g(t)-h(t) where g and h are the position functions of the two runners. Show that f,(c) = 0 for some c. You may assume that g and h are differentiable functions. Make sure you argue carefully why the conditions of the Mean Value Theorem are satisfied.

Explanation / Answer

The problem has already defined g(t), that is the position vector of first runner and h(t) be the position vector of second runner. Since the g(t) and h(t) are continuous and differentiable function, Reason: Because the distance will increase gradually (continuous), there won't be a sudden change since the human cannot vanish or something, hence g'(t) and h'(t) will be continuous

Let us assume the race gets completed in T time

f(0) = g(0) - h(0) = 0 [ since they both are starting point]

f(T) = g(T) - h(T) = 0 [since they both are at ending point]

Since f(t) is a difference of two continous and differentiable function, hence f(t) must also be continuous and differentiable

f'(t) = g'(t) - h'(t)

Using the mean's value theorem, if the function is continuous in [a,b] and differentiable in (a,b), then there exists a point c such that f'(c) = f(b) - f(a)/(b-a)

Now applying this thing for a=0, b=T

f'(c) = f(T) - f(0)/(T-0) = 0

Since f'(c) = 0, hence from our earlier calculation, it must be equal to g'(c) - h'(c)

g'(c) = h'(c) means that they have the same speed at the point c

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.