For (a) They tell you dVsphere/dt = KA = 4Kπr2. But you can also find dVsphere/dt by finding d/dt(4/3πr3). You'll wind up with a dr/dt term in that. You can then set the dV/dt equations equal, and rearrange to solve for dr/dt, plugging in 4 for r. You should get for part (a) that dr/dt is a constant number to answer (b). For part (c) find d2V/dt2 and see how that depends on d2r/dt2=0. If d2V/dt2 is 0, then the rate of change of the volume must be a constant, becuase the derivative of a constant is 0.
For the voltage problem: (a) Type the equation into a graphing calculator or computer program, where V(x)=y and t=x. (b) Find V'(t), and once you do, plug in 2 for t. (Also, is it supposed to be V(t)=1/t+tSin(t)?) Use the product rule for the tSin(t) term, and normal drop-down for the other. (c) The general principle behind a linear approximation is that you can get a ballpark idea on what the V(t) value is, the only catch being that you want to be somewhere near the value you're trying to approximate. So see if a line just touching the graph in part (a) near t=b is close to V(a) or V(b).
For (a) They tell you dVsphere/dt = KA = 4Kπr2. But you can also find dVsphere/dt by finding d/dt(4/3πr3). You'll wind up with a dr/dt term in that. You can then set the dV/dt equations equal, and rearrange to solve for dr/dt, plugging in 4 for r. You should get for part (a) that dr/dt is a constant number to answer (b). For part (c) find d2V/dt2 and see how that depends on d2r/dt2=0. If d2V/dt2 is 0, then the rate of change of the volume must be a constant, becuase the derivative of a constant is 0.
For the voltage problem:
(a) Type the equation into a graphing calculator or computer program, where V(x)=y and t=x.
(b) Find V'(t), and once you do, plug in 2 for t. (Also, is it supposed to be V(t)=1/t+tSin(t)?) Use the product rule for the tSin(t) term, and normal drop-down for the other.
(c) The general principle behind a linear approximation is that you can get a ballpark idea on what the V(t) value is, the only catch being that you want to be somewhere near the value you're trying to approximate. So see if a line just touching the graph in part (a) near t=b is close to V(a) or V(b).
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