Consider the function $$ f(x)=4 x^{2}-5 x-3 $$ and complete parts (a) through (c). (a) Find $$ f(a+h) $$. (b) Find $$ \frac{f(a+h)-f(a)}{h}, h
eq 0 $$. (c) Find the instantaneous rate of change of $$ f $$ when $$ a=7 $$.

Question

Consider the function $$ f(x)=4 x^{2}-5 x-3 $$ and complete parts (a) through (c). (a) Find $$ f(a+h) $$. (b) Find $$ \frac{f(a+h)-f(a)}{h}, h 
eq 0 $$. (c) Find the instantaneous rate of change of $$ f $$ when $$ a=7 $$.

UpStudy AI · Accepted Answer

Let's analyze the function $$ f(x) = 4x^2 - 5x - 3 $$ and address each part of the problem step by step.

### Part (a): Find $$ f(a+h) $$

To find $$ f(a+h) $$, substitute $$ x = a + h $$ into the original function:

$$
f(a+h) = 4(a + h)^2 - 5(a + h) - 3
$$

Let's expand and simplify this expression:

1. **Expand $$ (a + h)^2 $$:**
   $$
   (a + h)^2 = a^2 + 2ah + h^2
   $$

2. **Multiply by 4:**
   $$
   4(a + h)^2 = 4a^2 + 8ah + 4h^2
   $$

3. **Distribute the -5:**
   $$
   -5(a + h) = -5a - 5h
   $$

4. **Combine all terms:**
   $$
   f(a+h) = 4a^2 + 8ah + 4h^2 - 5a - 5h - 3
   $$

So, the expanded form of $$ f(a+h) $$ is:

$$
f(a+h) = 4a^2 + 8ah + 4h^2 - 5a - 5h - 3
$$

### Part (b): Find $$ \frac{f(a+h) - f(a)}{h} $$, where $$ h \neq 0 $$

First, calculate $$ f(a+h) - f(a) $$:

$$
f(a+h) - f(a) = (4a^2 + 8ah + 4h^2 - 5a - 5h - 3) - (4a^2 - 5a - 3)
$$

Simplify by subtracting $$ f(a) $$ from $$ f(a+h) $$:

$$
f(a+h) - f(a) = 8ah + 4h^2 - 5h
$$

Now, divide by $$ h $$:

$$
\frac{f(a+h) - f(a)}{h} = \frac{8ah + 4h^2 - 5h}{h} = 8a + 4h - 5
$$

So, the difference quotient is:

$$
\frac{f(a+h) - f(a)}{h} = 8a + 4h - 5
$$

### Part (c): Find the Instantaneous Rate of Change of $$ f $$ When $$ a = 7 $$

The instantaneous rate of change of $$ f $$ at $$ a = 7 $$ is the derivative of $$ f $$ at that point. From part (b), the derivative (as $$ h \to 0 $$) is:

$$
f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} = 8a - 5
$$

Substitute $$ a = 7 $$:

$$
f'(7) = 8(7) - 5 = 56 - 5 = 51
$$

**Answer Summary:**

**(a)** $$ f(a+h) = 4a^{2} + 8a\,h + 4h^{2} - 5a - 5h - 3 $$

**(b)** $$ \frac{f(a+h)-f(a)}{h} = 8a + 4h - 5 $$

**(c)** The instantaneous rate of change at $$ a = 7 $$ is 51.
**(a)** $$ f(a+h) = 4a^2 + 8ah + 4h^2 - 5a - 5h - 3 $$ **(b)** $$ \frac{f(a+h) - f(a)}{h} = 8a + 4h - 5 $$ **(c)** The instantaneous rate of change at $$ a = 7 $$ is 51.

Consider the function and complete parts (a) through ©.
(a) Find .
(b) Find .
© Find the instantaneous rate of change of when .

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Calculus Questions

Consider the function and complete parts (a) through ©. (a) Find . (b) Find . © Find the instantaneous rate of change of when .