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06:46Variable gravity At Earth’s surface, the acceleration due to gravity is approximately g=9.8 m/s² (with local variations). However, the acceleration decreases with distance from the surface according to Newton’s law of gravitation. At a distance of y meters from Earth’s surface, the acceleration is given by a(y) = - g / (1+y/R)², where R=6.4×10⁶ m is the radius of Earth.
b. Use the Chain Rule to show that dv/dt = 1/2 d/dy(v²).
Determine whether the following statements are true and give an explanation or counterexample.
b. The area of the region between y=sin x and y=cos x on the interval [0,π/2] is ∫π/20(cosx−sinx)dx.
Distance traveled and displacement Suppose an object moves along a line with velocity (in ft/s) v(t)=6−2t, for 0≤t≤6, where t is measured in seconds.
b. Find the displacement of the object on the interval 0≤t≤6.
40–43. Population growth
Starting with an initial value of P(0)=55, the population of a prairie dog community grows at a rate of P′(t)=20−t/5 (prairie dogs/month), for 0≤t≤200, where t is measured in months.
b. Find the population P(t), for 0≤t≤200.
Filling a tank A 2000-liter cistern is empty when water begins flowing into it (at t=0 at a rate (in L/min) given by Q′(t) = 3√t, where t is measured in minutes.
b. Find the function that gives the amount of water in the tank at any time t≥0.
55–58. Marginal cost Consider the following marginal cost functions.
b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units.
C′(x) = 300+10x−0.01x²
