Let fx 3 x2 5x Find a simplified formula for the average r

Let f(x) = 3 x^2 + 5x. Find a simplified formula for the average rate of change f(x + h0 - f(x)/h Suppose a colony of bacteria is known to grow exponentially with time. At the of 8 hours there are 7.000 bacteria At the end of 24 hours there are bacteria. How many bacteria were present initially? A tall monument cannot be measured by simply taking a long measuring tape and dropping it down from the top of the monument to the ground. Instead, trigonometry is used. For example, to measure the height of a lighthouse, a person stands away from the base and measures the angle of elevation to the top of the lighthouse to be 42 degree. Moving 98.6 feet closer, the angle of elevation to the top of the lighthouse is 78 degree How tall is the lighthouse?

Solution

13. f(x)= 3x²+5x

    f(x+h) = 3(x+h)²+5(x+h) = 3x²+3h²+6xh +5x+5h

Rate of change = (f(x+h) - f(x))/h = (3x²+3h²+6xh +5x+5h - 3x²-5x)/h

                                                 = (3h²+6xh+5h)/h = 3h + 6x+5

14. Exponential growth equation is

y=a b^x

At the end of 8 hours there are 7000 bacterias

Therefore x=8 and y=7000

7000=a b⁸

At the end of 24 hrs there are 54500 bacteria

Therefore x=24   y=56500

56500=a b²⁴

From these two equations

56500/7000= b¹⁶

b=(113/14)^1/16

7000= a(113/14)^8*1/16

7000= a(113/14)^.5

a= 7000/(113/14)^.5 = approx 2464

Therefore initially there are 2464 bacteria