Give step-by-step solution with explanation and final answer:PYRE
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— Se
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Define the function: -
ow = [ roa

Note: we're

Question

Give step-by-step solution with explanation and final answer:PYRE
3 A
-§-y 3-23 a GEC (1
-a
BE)
Ns — =
FELT)
— Se
~> 3 LL A busy (
eR
BR

Define the function: -
ow = [ roa

Note: we're using 7, not f, to define g(x). So you will want to use the graph

you made to answer these questions.

(2) Compute the following:
9(=5) = 9(=1) = 9(3) =
9(—4) = 9(0) = 9(3.5) =
9(-35) = 9(0.5) = 9(4) =
9(=3) = 9(1) = 9(5) =
9(=2) = 92) = 9(5.5) =
9(=15) = 9(2:5) = 9(6) =

(3) Now sketch the graph of y = g(z) using the points computed on the
previous page:
1

HESS ESSS SHEARS She HSE HR RHE EREHESTERD

BE de tea
(4) You graphed f(z) from differentiating f(z), and you graphed g(x) from
integrating f(z). In your own words, what do you observe between f(z) and
g(x). Be sure to offer your understanding reasoning around the relationship
between differentiation and integration. (To help, a graph of the original
function f(z) is shared below)

VIPSolutions · Accepted Answer

Let's break this down step by step using the information and graphs provided.

---

## **Step 1: Understanding the Definitions**

You are given:

$$ g(x) = \int_{-4}^{x} f'(t) \, dt $$

By the **Fundamental Theorem of Calculus**, integrating $ f'(t) $ from $-4$ to $x$ gives you $ f(x) - f(-4) $:

$$
g(x) = f(x) - f(-4)
$$
So $ g(x) $ is just the value of $ f(x) $ shifted vertically by $ -f(-4) $.

---

## **Step 2: Find $ f(-4) $ from the Graph**

From the graph of $ f(x) $:

- At $ x = -4 $, $ f(-4) = 2 $.

---

## **Step 3: Calculate $ g(x) $ for Each Value**

For each $ x $, use:

$$
g(x) = f(x) - 2
$$

Now, read off $ f(x) $ from the graph for each value and subtract 2.

---

### **Detailed Calculations**

#### **Left Column**
- $ g(-5) = f(-5) - 2 = 0 - 2 = -2 $
- $ g(-4) = f(-4) - 2 = 2 - 2 = 0 $
- $ g(-3.5) = f(-3.5) - 2 $  
  From the graph, $ f(-3.5) \approx 2.5 \Rightarrow 2.5 - 2 = 0.5 $
- $ g(-3) = f(-3) - 2 = 3 - 2 = 1 $
- $ g(-2) = f(-2) - 2 = 1 - 2 = -1 $
- $ g(-1.5) = f(-1.5) - 2 $  
  From the graph, $ f(-1.5) \approx 0 \Rightarrow 0 - 2 = -2 $

#### **Middle Column**
- $ g(-1) = f(-1) - 2 = -1 - 2 = -3 $
- $ g(0) = f(0) - 2 = 0 - 2 = -2 $
- $ g(0.5) = f(0.5) - 2 $  
  $ f(0.5) \approx 0.5 \Rightarrow 0.5 - 2 = -1.5 $
- $ g(1) = f(1) - 2 = 1 - 2 = -1 $
- $ g(2) = f(2) - 2 = 2 - 2 = 0 $
- $ g(2.5) = f(2.5) - 2 $  
  $ f(2.5) \approx 0.5 \Rightarrow 0.5 - 2 = -1.5 $

#### **Right Column**
- $ g(3) = f(3) - 2 = -1 - 2 = -3 $
- $ g(3.5) = f(3.5) - 2 $  
  $ f(3.5) \approx -2 \Rightarrow -2 - 2 = -4 $
- $ g(4) = f(4) - 2 = 0 - 2 = -2 $
- $ g(5) = f(5) - 2 = 1 - 2 = -1 $
- $ g(5.5) = f(5.5) - 2 $  
  $ f(5.5) \approx 1.5 \Rightarrow 1.5 - 2 = -0.5 $
- $ g(6) = f(6) - 2 = 1 - 2 = -1 $

---

### **Final Answers Table**

| $ x $      | $ g(x) $ |
|--------------|------------|
| -5           | -2         |
| -4           | 0          |
| -3.5         | 0.5        |
| -3           | 1          |
| -2           | -1         |
| -1.5         | -2         |
| -1           | -3         |
| 0            | -2         |
| 0.5          | -1.5       |
| 1            | -1         |
| 2            | 0          |
| 2.5          | -1.5       |
| 3            | -3         |
| 3.5          | -4         |
| 4            | -2         |
| 5            | -1         |
| 5.5          | -0.5       |
| 6            | -1         |

---

## **Step 4: Sketch the Graph of $ y = g(x) $**

- Plot the above points on the provided graph.
- Connect the points smoothly, noting that the shape of $ g(x) $ follows $ f(x) $ but shifted down by 2 units.

---

## **Step 5: Relationship Between $ f(x) $ and $ g(x) $**

### **Observation**

- $ g(x) $ is just $ f(x) $ shifted vertically downward by 2 units ($ f(-4) $).
- This illustrates the **Fundamental Theorem of Calculus**: Integrating the derivative $ f'(x) $ gives back the original function $ f(x) $, up to an additive constant (here, $ -f(-4) $).
- **Differentiation and integration are inverse processes** (up to a constant).

---

If you need the graph sketched or any further clarification, let me know!

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