A function is given. $$ f(t)=4 t^{2} ; \quad t=3, t=3+h $$ (a) Determine the net change between the given values of the variable. $$ \square $$ (b) Determine the average rate of change between the given values of the variable. $$ \square $$

Question

UpStudy AI · Accepted Answer

**(a) Net Change**

1. Evaluate the function at $$ t = 3+h $$:
   $$
   f(3+h) = 4(3+h)^2
   $$

2. Expand $$(3+h)^2$$:
   $$
   (3+h)^2 = 9 + 6h + h^2
   $$

3. Thus,
   $$
   f(3+h) = 4(9 + 6h + h^2) = 36 + 24h + 4h^2
   $$

4. Evaluate the function at $$ t = 3 $$:
   $$
   f(3) = 4 \times 3^2 = 36
   $$

5. The net change is:
   $$
   \Delta f = f(3+h) - f(3) = (36 + 24h + 4h^2) - 36 = 24h + 4h^2
   $$

**(b) Average Rate of Change**

1. The average rate of change between $$ t = 3 $$ and $$ t = 3+h $$ is defined as:
   $$
   \text{Average Rate of Change} = \frac{f(3+h) - f(3)}{(3+h) - 3}
   $$

2. Substitute the values obtained:
   $$
   \frac{24h + 4h^2}{h}
   $$

3. Factor the numerator:
   $$
   \frac{4h(6 + h)}{h}
   $$

4. Simplify (assuming $$ h \neq 0 $$):
   $$
   4(6+h) = 24 + 4h
   $$
**(a) Net Change:** $$ 24h + 4h^2 $$ **(b) Average Rate of Change:** $$ 24 + 4h $$

A function is given. \[ f(t)=4 t^{2} ; \quad t=3, t=3+h \] (a) Determine the net change between the given values of the variable. \( \square \) (b) Determine the average rate of change between the given values of the variable. \( \square \)

Upstudy AI Solution

Answer

Solution

Mind Expander

Related Questions

Latest Calculus Questions