Motion Along a Line In Exercises 81-84, the function
describes the motion of a particle along a line. (a) Find the velocity function of the particle at any time
. (b) Identify the time interval (s) on which the particle is moving in a positive direction. (c) Identify the time interval(s) on which the particle is moving in a negative direction, (d) Identify the time(s) at which the particle changes direction.
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Chapter 3 Solutions
Calculus (MindTap Course List)
- Hooke’s Law In Exercises 65-68, use Hooke’s Law, which states that the distance a spring stretches (or compresses) from its natural, or equilibrium, length varies directly as the applied force on the spring. The coiled spring of a toy supports the weight of a child. The weight of a 25-pound child compresses the spring a distance of 1.9 inches. The toy does not work properly when a weight compresses the spring more than 3 inches. What is the maximum weight for which the toy works properly?arrow_forwardFreight on Class I Railroads According to the Association of American Railroads, Class I freight railroads are the line-haul freight railroads with 2006 operating revenue in excess of 346.8million. Let F=F(t) denote the freight revenue in billions of dollars of Class I railroads in year t. In 2005, Class I railroads had a freight revenue of 44.5billion. In 2007, the revenue was 52.9 billion. Calculate the average rate of change per year in F from 2005 to 2007 and explain its meaning in practical terms.arrow_forwardA bicycle is classified by the diameter of its tires. For example, a 20-inch bicycle has tires that are 20 inches in diameter. The number of times a bicycle tire rotates in a given period of time is directly related to the distance traveled in that period of time. A 26-inch bicycle is pedaled so that the tires rotate at a rate of 200 revolutions per minute. A 20-inch bicycle is pedaled so that its tires rotate at a rate of 200 revolutions per minute, as well. How many complete rotations does either bike complete in one second? Answer in a complete sentence.arrow_forward
- A bicycle is classified by the diameter of its tires. For example, a 20-inch bicycle has tires that are 20 inches in diameter. The number of times a bicycle tire rotates in a given period of time is directly related to the distance traveled in that period of time. A 26-inch bicycle is pedaled so that the tires rotate at a rate of 200 revolutions per minute. A 20-inch bicycle is pedaled so that its tires rotate at a rate of 200 revolutions per minute, as well. What is the linear velocity of each bicycle? State your answers in inches per minute rounded to the nearest tenth.arrow_forwardCalculus Find the velocity, acceleration, and speed of a particle with the given position function.arrow_forwardDetermining Differentiability In Exercises85–88, find the derivatives from the left and fromthe right at x = 1 (if they exist). Is the functiondifferentiable at x =1? \text { 8. } f(x)=(1-x)^{2 / 3}arrow_forward
- The rate of disbursement dQ/dt of a $8 million federal grant is proportional to the square of 100 − t, where t is the time (in days, 0 ≤ t ≤ 100) and Q is the amount that remains to be disbursed. Find the amount that remains to be disbursed after 50 days. Assume that the entire grant will be disbursed after 100 days.arrow_forwardSlope of a Curve at a PointIn Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.arrow_forwardA body of mass m fell from a vertical plane. Find the velocity of the object as a function of time t. Assume that the air resistance is proportional to the velocity of the objectarrow_forward
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