Rate of Change for Ball Thrown: How to Use the Formula f(b)-f(a)/b-a

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SUMMARY

The discussion focuses on calculating the rate of change for a ball thrown, utilizing the formula (f(b) - f(a)) / (b - a). The function h(t) is defined as h(t) = 56t - 16t², which simplifies to h(t) = -16(t - 7/2)² + 196 after completing the square. Participants emphasize the importance of correctly applying parentheses in the formula to avoid misinterpretation. The average rate of change is defined as total displacement over the time interval.

PREREQUISITES
  • Understanding of basic calculus concepts, specifically rate of change
  • Familiarity with quadratic functions and their properties
  • Knowledge of completing the square technique in algebra
  • Ability to manipulate algebraic expressions and fractions
NEXT STEPS
  • Practice calculating rates of change using various functions
  • Explore the implications of the average rate of change in real-world scenarios
  • Learn more about the properties of quadratic functions and their graphs
  • Study the significance of parentheses in mathematical expressions to avoid errors
USEFUL FOR

Students studying calculus, educators teaching mathematical concepts, and anyone interested in understanding the application of rate of change in physics and mathematics.

Niaboc67
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Homework Statement


10pdx1D.png

The Attempt at a Solution


As far as I know those are correct. Could I have some assistance on how to solve for the rate of change? I know the formula is f(b)-f(a)/b-a but not sure how to go about it
 
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Niaboc67 said:

Homework Statement

The Attempt at a Solution


As far as I know those are correct. Could I have some assistance on how to solve for the rate of change? I know the formula is f(b)-f(a)/b-a but not sure how to go about it

What if f(t) = h(t)? Could you pick two times, t1 = a and t2 = b, and calculate the rate of change, knowing h(t) as a function of t?
 
Niaboc67 said:

Homework Statement


10pdx1D.png

The Attempt at a Solution


As far as I know those are correct. Could I have some assistance on how to solve for the rate of change? I know the formula is f(b)-f(a)/b-a but not sure how to go about it

That formula is saying take the change in height f(b) - f(a) and divide by the change in time b - a. Essentially, avg rate = total displavement over time for that displacement
 
Unfortunately most of the answers you give here are wrong. You are given that h(t)= 56t- 16t^2= -16(t^2- (7/2)t). Completing the square gives h(t)= -16(t- 7/2)^2+ 196.
 
Niaboc67 said:
I know the formula is f(b)-f(a)/b-a but not sure how to go about it
Use parentheses!
What you wrote is the same as ##f(b) - \frac{f(a)}{b} - a##.

If you write a fraction such as the one above in single-line text, it has to be written as (f(b) - f(a))/(b - a).
 

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