For the periodic function \(f(x)=4\sin(\pi x)+1\), the period is the amplitude is the average value is the maximum value is the minimum value is Desmos (click to reveal)

If \(\displaystyle\lim_{x\to 2}f(x)=\) `f` and \(\displaystyle\lim_{x\to 2}g(x)=\) `g`, what is the value of \(\displaystyle\lim_{x\to 2}\) (`a` \(f(x)+\)`b`\(g(x)\))?

The graph below shows the function \(f(x)\) along with a dashed line. The line in the graph is a line. The slope of this line is an rate of change of \(f(x)\).

The metabolism of a \(t\)-day-old albatross chick is given by \(m(t)\) watts per kilogram (\(\tfrac{W}{kg}\)).  \(m\,^\prime(10)=2\) means that when an albatross chick is old, its metabolism is . \(m(10)=4\) means that when an albatross chick is old, its metabolism is .    

Use the graph of \(f(x)\) to fill in the blanks. If a value does not exist, type DNE.   \(\displaystyle\lim_{x\to 4^-}f(x)=\) \(\displaystyle\lim_{x\to 4^+}f(x)=\) \(\displaystyle\lim_{x\to 4}f(x)=\) \(f(4)=\) Is \(f(x)\) continuous at \(x=4\)?

Numerically estimate \(\displaystyle\lim_{x\to 3^-}\left(1.25^x-0.1x\right),\) accurate to two decimal places. Fill in the table according to the conventions we have used in the textbook, practice problems, and homework.   \(x\to 3^-\) \(1.25^x-0.1x\) 2.9 We conclude that \(\displaystyle\lim_{x\to 3^-}\left(1.25^x-0.1x\right)=\)   Desmos (click to reveal)

Use the information below to answer the two (2) following questions:An inventor develops a stationary cycling device by which an individual, while pedaling, can convert all of the energy expended into heat for warming water. How much mechanical energy is required to increase the temperature of 300 g of water (enough for 1 cup of coffee) from 20°C to 95°C? (1 cal = 4.186 J, the specific heat of water is 4 186 J/kg⋅°C)

Use the information below to answer the four (4) following questions: A piston containing 1.0 mole of an ideal monatomic gas undergoes the following cycle: (i.) isochoric heating increasing pressure from 2 kPa to 5 kPa (ii.) isobaric expansion from 2 m3 to 8 m3(iii.) isochoric cooling decreasing pressure from 5 kPa to 2 kPa(iv.) isobaric contraction from 8 m3 to 2 m3Remember:  R = 8.314 J/mol K, cv mol = 12.47 J/mol K Calculate the temperature of the gas when it is at the point labeled a on the PV diagram.

Use the information below to answer the three (3) following questions: A cylindrical grindstone of mass m = 18 kg and radius R = 0.8 m is initially rotating freely at 26π radians/s.  An axe is brought into contact with the grindstone which brings it to a stop after 30 revolutions. What is the magnitude of the angular acceleration of the grindstone?

In the figure, R1 = R2 and cm represents the center of mass of the object.  The rotational inertia about an axis through point P1 is I1, the rotational inertia about an axis through point P2 is I2, and the rotational inertia about an axis through the cm is Icm.  The relationship among the moments is