Homework 7 Implicit differentiation Exercises

Exercise 7.1: In each case use the tangent line approximation to estimate the y-coordinate of the point on the curve whose horizontal distance is 0.01 (to the right) from the given point P:

(a) , P = (1,2);

\vfill

(b) , P = (2,5);

\vfill

(c) , P = (9,4).

\vfill\newpage

Exercise 7.2. Find a formula for the derivative at the point ( ):

(a) ;

\vfill

(b) ;

\vfill

(c) .

\vfill\newpage

Exercise 7.3: A hyperbola passing through ( ) consists of all points whose distance
from the origin is a constant more than its distance from the point ( ). Find the slope of the tangent line to the hyperbola at ( ).
\vfill

Exercise 7.4: In Example 7.5 (a), suppose that you are starting from the point (1,1) rather than (0,1). Find the equation of the line from this point which is the closest approximation to the curve of constant total light intensity. After you've walked a horizontal distance of 0.1 (to the right), what is your y-coordinate?
\fill\newpage

Exercise 7.5 : In the situation of Example 7.5, suppose that you don't know the relative strengths of the sources A and B, but you know that the line is the line that you should start walking along from the point ( ) if you want the total light intensity to be as constant as possible. Find how much stronger B is than A.
\vfill

Exercise 7.6: Suppose that a light source A is at the origin, and a source B twice as strong is located at ( ). Find the slope of the direction you move in from ( ) in order for the total light intensity to remain as constant as possible.
\vfill\newpage

Exercise 7.7: When crude oil flows from a well, water is frequently mixed with it in an emulsion. To remove the water the crude is piped to a device called a heater--treater, which is simply a large tank in which the oil is warmed and the water is allowed to settle out. Operating experience in a particular oil field indicates that the concentration of water in the treater's output can be modeled by the following equation in the neighborhood of the usual operating point of F and a 2-hour holding time:

where h is the holding time in hours and T is the operating temperature in degrees F. (a) Due to random fluctuations in the well's flow rate, the holding time actually varies slightly around 2 hours. Suppose you are given a simple control device that can change the tank temperature proportionally to the measured change in holding time. What constant of proportionality would best compensate for small holding time fluctuations and keep the water concentration as constant as possible? (b) Now as field equipment ages, its maximum operating settings are generally decreased. Find the equation of the line that best approximates the way in which the holding time would have to be increased as the maximum temperature rating falls slightly below the usual operating temperature.

\vfill

Exercise 7.8: One end of a string (the point A) is attached to the y-axis at a distance 20 meters from the origin; the string is threaded through a hole (the point B) at the end of a pole situated along the line ; and the other end of the string (the point C) is attached to a hook that is free to slide along the x-axis and is being pulled by a force to the right that is sufficient to keep the string taut. See the diagram. At time , when the top of the pole is at the point ( ), the pole starts lowering at the steady rate of ( = ). The sliding end of the string reaches the point at time .

code for graph