<< Chapter < Page Chapter >> Page >

Rate of change of temperature

A homeowner sets the thermostat so that the temperature in the house begins to drop from 70 ° F at 9 p.m., reaches a low of 60 ° during the night, and rises back to 70 ° by 7 a.m. the next morning. Suppose that the temperature in the house is given by T ( t ) = 0.4 t 2 4 t + 70 for 0 t 10 , where t is the number of hours past 9 p.m. Find the instantaneous rate of change of the temperature at midnight.

Since midnight is 3 hours past 9 p.m., we want to compute T ( 3 ) . Refer to [link] .

T ( 3 ) = lim t 3 T ( t ) T ( 3 ) t 3 Apply the definition. = lim t 3 0.4 t 2 4 t + 70 61.6 t 3 Substitute T ( t ) = 0.4 t 2 4 t + 70 and T ( 3 ) = 61.6 . = lim t 3 0.4 t 2 4 t + 8.4 t 3 Simplify. = lim t 3 0.4 ( t 3 ) ( t 7 ) t 3 = lim t 3 0.4 ( t 3 ) ( t 7 ) t 3 = lim t 3 0.4 ( t 7 ) Cancel. = −1.6 Evaluate the limit.

The instantaneous rate of change of the temperature at midnight is −1.6 ° F per hour.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Rate of change of profit

A toy company can sell x electronic gaming systems at a price of p = −0.01 x + 400 dollars per gaming system. The cost of manufacturing x systems is given by C ( x ) = 100 x + 10,000 dollars. Find the rate of change of profit when 10,000 games are produced. Should the toy company increase or decrease production?

The profit P ( x ) earned by producing x gaming systems is R ( x ) C ( x ) , where R ( x ) is the revenue obtained from the sale of x games. Since the company can sell x games at p = −0.01 x + 400 per game,

R ( x ) = x p = x ( −0.01 x + 400 ) = −0.01 x 2 + 400 x .

Consequently,

P ( x ) = −0.01 x 2 + 300 x 10,000 .

Therefore, evaluating the rate of change of profit gives

P ( 10000 ) = lim x 10000 P ( x ) P ( 10000 ) x 10000 = lim x 10000 −0.01 x 2 + 300 x 10000 1990000 x 10000 = lim x 10000 −0.01 x 2 + 300 x 2000000 x 10000 = 100 .

Since the rate of change of profit P ( 10,000 ) > 0 and P ( 10,000 ) > 0 , the company should increase production.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

A coffee shop determines that the daily profit on scones obtained by charging s dollars per scone is P ( s ) = −20 s 2 + 150 s 10 . The coffee shop currently charges $ 3.25 per scone. Find P ( 3.25 ) , the rate of change of profit when the price is $ 3.25 and decide whether or not the coffee shop should consider raising or lowering its prices on scones.

P ( 3.25 ) = 20 > 0 ; raise prices

Got questions? Get instant answers now!

Key concepts

  • The slope of the tangent line to a curve measures the instantaneous rate of change of a curve. We can calculate it by finding the limit of the difference quotient or the difference quotient with increment h .
  • The derivative of a function f ( x ) at a value a is found using either of the definitions for the slope of the tangent line.
  • Velocity is the rate of change of position. As such, the velocity v ( t ) at time t is the derivative of the position s ( t ) at time t . Average velocity is given by
    v ave = s ( t ) s ( a ) t a .

    Instantaneous velocity is given by
    v ( a ) = s ( a ) = lim t a s ( t ) s ( a ) t a .
  • We may estimate a derivative by using a table of values.

Key equations

  • Difference quotient
    Q = f ( x ) f ( a ) x a
  • Difference quotient with increment h
    Q = f ( a + h ) f ( a ) a + h a = f ( a + h ) f ( a ) h
  • Slope of tangent line
    m tan = lim x a f ( x ) f ( a ) x a
    m tan = lim h 0 f ( a + h ) f ( a ) h
  • Derivative of f ( x ) at a
    f ( a ) = lim x a f ( x ) f ( a ) x a
    f ( a ) = lim h 0 f ( a + h ) f ( a ) h
  • Average velocity
    v a ve = s ( t ) s ( a ) t a
  • Instantaneous velocity
    v ( a ) = s ( a ) = lim t a s ( t ) s ( a ) t a

For the following exercises, use [link] to find the slope of the secant line between the values x 1 and x 2 for each function y = f ( x ) .

Practice Key Terms 4

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?

Ask