Logarithmic Differentiation 对数求导

Example 16

Example 17

3.7 Higher Derivatives 高阶导数

Example 18

Example 19

Solution Example 20

3.10 Related Rates 相关变化率

Related Rates  In a related-rates problem, the idea is to compute the rate of change of one quantity in terms of the rate of change of another quantity—which may be more easily measued. （在相关变化率的问题中，通常是想要利用某个量的 变化率来求另一个量的变化率）  The procedure is to find an equation that relates the two quantities and then use the Chain Rule to differentiate both sides with respect to time. （做法是先找到描述两个量之间关系的方程式，然后 利用链式法则将等号两边对时间做微分）

Example 1 Air is being pumped into a spherical balloon so that its volume increases at a rate of 100cm 3 /s. How fast is the radius of the balloon increasing when the diameter is 50cm? Solution Let V be the volume of the balloon and let r be its radius. The volume and the radius are both functions of time t. The rate of increase of the volume with respect to time is dV/dt,and the rate of increase of the radius is dr/dt.

In order to connect dV/dt and dr/dt, we first relate V and r by the formula for the volume of a sphere: Differentiate each side of this equation with respect to t Put r=25 and dV/dr=100 in this equation, we obtain:

Example 2 A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall? Solution Let x feet be the distance from the bottom of the ladder to the wall and y feet the distance from the top of the ladder to the ground. Note x and y are both function of t.

The relationship between x and y are When x=6, the Pythagorean Theorem gives y=8, so we obtain: Differentiate each side of this equation with respect to t

Example 3 A water tank has the shape of an inverted circular cone with base radius 2 m and heihgt 4 m. If water is being pumped into the tank at a rate of 2 m 3 /min, find the rate at which the water level is rising when the water is 3 m deep. Solution Let V,r and h be the volume of the water, the radius of the surface, and the height. Note that V and h are function with respect to time t.

The quantities V and h are related by the equation Thus we have: From the figure we know: When h=3 m and dV/dt=2 m 3 /min:

Solution Let C be the intersection of the roads. At a given time t, let x be the distance from car A to C, let y be the distance from car B to C, and let z be the distance between the cars, where x,y,z are measured in miles.

The equation that relates x,y, and z is given by the Pythagorean Theorem: Thus we have: When x=0.3 mi and y=0.4 mi, the Pythagorean Theorem gives z=0.5 mi:

3.11 Linear Approximations and Differentials 线性逼近和微分

 A curve y=f (x) lies very close to its tangent line near the point tangency.  We use the tangent line at (a,f (a)) as an approximation to the curve y=f (x) when x near to a.

 The tangent line at a is  It is called the linear approximation( 线性逼近） or tangent line approximation （切线逼近） of f at a.

Solution Let T(t) is the temperature of the turkey after t hours. So Example 1 Suppose that after you stuff a turkey its temperature is 50 0 F and you then put it in a F oven. After an hour the meat thermometer indicates that the temperature of the turkey is 93 0 F and after two hours it indicates F. Predict the temperature of the turkey after three hours.

The temperature of the turkey after three hours

Solution Since The linearization is: So :

Our approximations are overestimates because the tangent line lies above the curve. This approximations give good estimates when x is close to 1, but the accuracy of the approximation deteriorate when x is far away from 1.

Differentials( 微分）