A straight highway connects towns [math]A[/math] and [math]B[/math]. A car starts at [math]t = 0[/math] from [math]A[/math] and goes toward [math]B[/math] with a velocity given by [math]v(t) = 60t - 12t^2[/math], measured in miles per hour. When the car arrives at [math]B[/math], it is slowing down and its speed is [math]48[/math] miles per hour.

• How far apart are the two towns?
• What are the maximum and minimum speeds obtained during the trip? When are they reached, and at what distances?

A straight highway connects towns [math]A[/math] and [math]B[/math]. A car, initially stopped, starts at [math]t = 0[/math] from [math]A[/math] and accelerates at [math]240[/math] miles per hour per hour until reaching a speed of [math]60[/math] miles per hour.

• How long does this take, both in time and distance? Assume that the car travels at the constant speed of [math]60[/math] miles per hour once it has reached that speed, and that is slows down to a stop at town [math]B[/math] in the same way that it left [math]A[/math].
• How far apart are [math]A[/math] and [math]B[/math] if the whole trip takes [math]5[/math] hours?

A projectile is fired straight up with an initial velocity of [math]640[/math] feet per second (see Example \ref{exam 4.8.2}).

• Find the velocity [math]v(t)[/math].
• How far does the projectile travel during the first [math]10[/math] seconds of its flight?
• How far does the projectile go, and how many seconds after takeoff is this maximum height reached?
• What is the total distance traveled by the projectile during the first [math]30[/math] seconds of its flight?
• What is the velocity when the projectile returns to the ground?

Let the function [math]f[/math] be integrable over the interval [math][a,b][/math], and suppose that [math]f(x)[/math] does not change sign on the interval. Prove that

(This is an easy problem. Consider separately the two cases: First, [math]f(x) \geq 0[/math] for every [math]x[/math] in [math][a,b][/math], and second, [math]f(x) \leq 0[/math] for every [math]x[/math] in [math][a,b][/math].)

A particle moves on the [math]x[/math]-axis with velocity given by [math]v(t) = -4t + 20[/math].

• In which direction is the particle moving at time [math]t = 0[/math]?
• Find [math]s(t)[/math], the position of the particle at time [math]t[/math], if its coordinate is [math]-30[/math] when [math]t = 1[/math].
• Find the distance traveled by the particle during the time interval from [math]t = 0[/math] to [math]t = 4[/math].
• Find the distance traveled by the particle during the time interval from [math]t = 0[/math] to [math]t = 8[/math].
• When is [math]s(t) = 0[/math]?

A particle moves on the [math]y[/math]-axis with acceleration given by [math]a(t) = 6t - 2[/math]. Denote its velocity and position at time [math]t[/math] by [math]v(t)[/math] and [math]y(t)[/math], respectively. At time [math]t = 1[/math], the particle is at rest at the zero position.

• Find [math]v(t)[/math] and [math]y(t)[/math].
• How far does the particle move during the time interval from [math]t = 1[/math] to [math]t = 3[/math]?
• What is the distance traveled by the particle from [math]t = -1[/math] to [math]t = 2[/math]?

A road borders a rectangular forest, and a car is driven around it. The car starts from rest at one corner and accelerates at [math]120[/math] miles per hour per hour until it reaches the next corner [math]15[/math] minutes later. The second side is [math]20[/math] miles long and the car is driven along it at constant velocity equal to the final velocity reached on the first side. The car continues at this same speed on the third side. On the fourth side, however, the car slows down with constant acceleration and comes to a stop at its original starting place. Find

• the dimensions of the rectangle.
• the acceleration on the fourth side.
• the time taken for the whole trip.

Let the function [math]f[/math] be integrable over the interval [math][a,b][/math]. From the definition of integrability in Section \secref{4.1}, prove that [math]\int_a^b f[/math] is the only number such that

for every partition [math]\sigma[/math] of [math][a,b][/math].

A conical funnel of height [math]36[/math] inches and base with radius [math]12[/math] inches is initially filled with sand. At [math]t = 0[/math], the sand starts running out the bottom (apex of the cone) so that the volume [math]V[/math] of sand remaining in the funnel is decreasing at the constant rate of [math]10[/math] cubic inches per minute.

• Find [math]V[/math] as a function of time [math]t[/math], and determine how long it takes for all the sand to run out.
• Assuming that the sand retains its original conical shape during the process, find the radius [math]r[/math] of the base of the cone of sand as a function of [math]t[/math].

A particle moves on the parabola [math]y = x^2[/math], and its horizontal component of velocity is given by [math]x^\prime (t) = \frac1{(t + 1)^2}[/math], [math]t \geq 0[/math]. At time [math]t = 0[/math] the particle is at the origin.

• What are the [math]x[/math] and [math]y[/math] coordinates of the particle when [math]t = 1[/math]? When [math]t = 3[/math]?
• As [math]t[/math] increases without bound what happens to the particle?