MTH 251 – Differential Calculus Assignment

Product and Quotient Rules

1. Find the derivatives of the following functions.

2.Find the derivative of f (x) and then determine the equation of the tangent line to f at x = 0.

Rates of Change

1. The atmospheric CO₂ level A(t) at Mauna Loa, hawaii, at time t (in parts per million by volume) is recorded by the Scripps Institution of Oceanography. Reading across, the values for the 4-year intervals are

Estimate A ' (t) in 1962, 1970, 1986, and 2002. In which of the years did the approximation take on its largest and smallest values? In which of these years does the approximation suggest that the CO₂ level was the most constant?

2. The dollar cost of producing x video cameras is C(x) = 500x ? 0.003x ² + 10^?8 x ³ . Estimate the marginal cost at production level x = 5000 and compare it with the actual marginal cost C(5001) ? C(5000). Compare this marginal cost with the average cost per camera, defined as C(x)/x.

3. A particle moving along a line has position s(t) = t 4 ? 18t ² meters at time t seconds. At which times does the particle pass through the origin? At which times is the particle instantaneously motionless (zero velocity

- Higher Derivatives

1. Find a general formula for f (n) (x) if f (x) = x ² e ^x .

2. Find the acceleration at time t = 5 min of a helicopter whose height is s(t) = 300t ? 4t ³ meters. Plot the acceleration s 00 for 0 ? t ? 6. Is the helicopter speeding up or slowing down during this time interval? Explain