Instantaneous Rate of Change at a Point - Page 10

Questions
  • I am not familiar with "increasing at a point". I know that #f(x) = x^3# is increasing on the whole real line, but since #f'(0)=0# do some people say that it is not increasing at #x = 0#?
  • If a cup of coffee has temperature 95∘C in a room where the temperature is 20∘C, then, according to Newton's Law of Cooling, the temperature of the coffee after t minutes is T(x)=20+75e^(-t/50). ?
  • The instantaneous growth rate r of a colony of bacteria t hours after the start of an experiment is given by the function #r(t)=0.01t^3-0.07t^2+ 0.07t +0.15#. What are the times for which the instantaneous growth rate is zero?
  • Two sides of a triangle are #6m# and #7m# in length and the angle between them is increasing at a rate of #0.06# rad/s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is #π/3# rad?
  • What is instantaneous rate of change as the limit of average rate of change?
  • Can you check the follow statements about derivatives?
  • What’s the correct way to solve this problem? Explain step by step. Thank you.
  • A new born child grows at a rate of #4/sqrtt# pounds per month for the first 12 months (#t>0#). How many pounds does the child gain between the 1st and 4th month?
  • If f(x) = 3x^2 + 8x, the slope of the tangent to f(x) when x=5 is 38 and the instantaneous rate of change for f(x) is also 38, what would the equation of the line tangent to f(x) when x=5 be?
  • A circular balloon is inflated with air flowing at a rate of 10cm^3/s. How fast is the radius of the balloon increasing when the radius is: 1cm? 10cm? 100cm?
  • Water flows from the bottom of a storage tank at a rate of r(t) = 210 - 5t liters per minute?
  • If f(x) = 5 x + 8, find the instantaneous rate of change of f( x ) at x=-7 ?
  • Use the difference quotient to estimate the instantaneous rate of change in f(x)=#x^2# at x=3?
  • How do you find the derivative of #4x^3-3x+8# by first principles?
  • Use the difference quotient to estimate the instantaneous rate of change in f(x)=#x^2# at x=3?
  • The volume of a spherical balloon is increasing at a constant rate of #0.25m^3s^-1#. Find the rate at which the radius is increasing at the instant when the volume is #10m^3#?
  • Estimate the instantaneous rate of change in the area of a circle when the radius is 3cm? The formula for the area of a circle is A=#pir^2#.