A freight train consists of two $8.00 \times 10^{4}-\mathrm{kg}$ engines and 45 cars with average masses of $5.50 \times 10^{4}\ \mathrm{kg}.$ (a) What force must each engine exert backward on the track to accelerate the train at a rate of $5.00 \times 10^{-2}\ \mathrm{m} / \mathrm{s}^{2}$ if the force of friction is $7.50 \times 10^{5}\ \mathrm{N},$ assuming the engines exert identical forces? This is not a large frictional force for such a massive system. Rolling friction for trains is small, and consequently trains are very energy-efficient transportation systems. (b) What is the force in the coupling between the 37th and 38th cars (this is the force each exerts on the other) assuming all cars have the same mass and that friction is evenly distributed among all of the cars and engines?

A freight train consists of two $8.00 \times 10^{5}-\mathrm{kg}$ engines and 45 cars with average masses of $5.50 \times 10^{5}\ \mathrm{kg}.$ (a) What force must each engine exert backward on the track to accelerate the train at a rate of $5.00 \times 10^{-2}\ \mathrm{m} / \mathrm{s}^{2}$ if the force of friction is $7.50 \times 10^{5}\ \mathrm{N},$ assuming the engines exert identical forces? This is not a large frictional force for such a massive system. Rolling friction for trains is small, and consequently trains are very energy-efficient transportation systems. (b) What is the magnitude of the force in the coupling between the 37th and 38th cars (this is the force each exerts on the other), assuming all cars have the same mass and that friction is evenly distributed among all of the cars and engines?

The increase in carbon dioxide $(CO_{2})$ in the atmosphere is a major cause of global warming. Using data obtained by Charles David Keeling, professor at Scripps Institution of Oceanography, the average amount of $CO_{2}$ in the atmosphere from 1958 through 2016 is approximated by $A(t)=0.012414 t^{2}+0.7485 t+313.9 \ (1 \leq t \leq 59)$ where A(t) is measured in parts per million volume (ppmv) and t in years, with t = 1 corresponding to 1958 Find the average rate of increase of the average amount of $CO_{2}$ in the atmosphere from 1958 through 2016.

Martha McDonald, a candidate for city council, must decide whether to come out in favor of a new factory, be opposed to it, or waffle on the issue. The change in votes for Martha depends on what her opponent, Kevin Wallace, does, with payoffs as shown.

$$\begin{matrix} \text{ } & \text{ } & \text{ } & \text{Kevin} & \text{ }\\ \text{ } & \text{ } & \text{Favors} & \text{Waffles} & \text{Opposes}\\ \text{ } & \text{Favors} & \text{0} & \text{1200} & \text{4200}\\ \text{Martha} & \text{Waffles} & \text{-1000} & \text{0} & \text{600}\\ \text{ } & \text{Opposes} & \text{-5000} & \text{-2000} & \text{0}\\ \end{matrix}$$

Find the optimum strategy for Martha and her opponent, and find the value of the game.

What force does a trampoline have to apply to Jennifer, a 45.0-kg gymnast, to accelerate her straight up at 7.50 m/s²? The answer is independent of the velocity of the gymnast - she can be moving up or down or can be instantly stationary.

A cleaner pushes a 4.50-kg laundry cart in such a way that the net external force on it is 60.0 N. Calculate the magnitude of his cart's acceleration.

A cleaner pushes a 4.50-kg laundry cart in such a way that the net external force on it is 60.0 N. Calculate the magnitude of its acceleration.

(a) Calculate the tension in a vertical strand of spider web if a spider of mass $8.00 \times 10^{-5} \mathrm{~kg}$ hangs motionless on it. (b) Calculate the tension in a horizontal strand of spider web if the same spider sits motionless in the middle of it much like the tightrope walker in given figure. The strand sags at an angle of $12^{\circ}$ below the horizontal. Compare this with the tension in the vertical strand (find their ratio).

Can a goalkeeper at her/ his goal kick a soccer ball into the opponent’s goal without the ball touching the ground? The distance will be about 95 m. A goalkeeper can give the ball a speed of 30 m/s.

The year-end inflation rates in the U.S. economy in 2009-2012 were 2.7%, 1.5%, 3.0%, and 1.7%, respectively. What was the purchasing power of a dollar at the beginning of 2013 compared to that at the beginning of 2009?

For the following exercise, find $\frac{d y}{d x}$ for the given function. $y=\cos ^{-1}(2 x) \cdot \sin ^{-1}(2 x)$

A ball is kicked with an initial velocity of 16 m/s in the horizontal direction and 12 m/s in the vertical direction. (a) At what speed does the ball hit the ground? (b) For how long does the ball remain in the air? (c) What maximum height is attained by the ball?

Two teams of nine members each engage in tug-of-war. Each of the first team's members has an average mass of 68 kg and exerts an average force of 1350 N horizontally. Each of the second team's members has an average mass of 73 kg and exerts an average force of 1365 N horizontally. (a) What is magnitude of the acceleration of the two teams, and which team wins? (b) What is the tension in the section of rope between the teams?

Let f and g be differentiable functions of x. Then

$$\begin{array}{ll} (c f)^{\prime}=c f^{\prime} & (f \pm g)^{\prime}=f^{\prime} \pm g^{\prime} \\ (f g)^{\prime}=f^{\prime} g+f g^{\prime} & \left(\frac{f}{g}\right)^{\prime}=\frac{f^{\prime} g-f g^{\prime}}{g^{2}}. \end{array}$$

Also, the chain rule is essential. $(f \circ g)^{\prime}(x)=f^{\prime}(g(x)) g^{\prime}(x)$ Use these rules, plus the table of derivatives in Table 1, to find the derivative of each of the functions.

$$\begin{array}{cc} \text { Table 1 } & \begin{array}{c} \text { A table of } \\ \text { derivatives } \end{array} \\ \hline f(x)= & f^{\prime}(x)= \\ \hline C & 0 \\ x & 1 \\ x^{n} & n x^{n-1} \\ \cos (x) & -\sin (x) \\ \sin (x) & \cos (x) \\ e^{x} & e^{x} \\ \ln (|x|) & 1 / x \end{array}$$

$f(x)=\frac{x^{2}}{\ln x}$