Math  /  Calculus

QuestionUse the graph of f(x)f(x) to evaluate the following:
The average rate of change of ff from x=1x=1 to x=4x=4 is \square Give your answer as an integer or reduced fraction. Question Help: Video

Studdy Solution

STEP 1

What is this asking? Find how much, on average, the function's output changes per unit change in the input between x=1x = 1 and x=4x = 4. Watch out! Don't mix up the points!
Make sure you're using the right yy values with the right xx values.

STEP 2

1. Identify the coordinates
2. Calculate the average rate of change

STEP 3

Alright, let's **locate** our points!
We're looking at the graph between x=1x = 1 and x=4x = 4.
The graph tells us that when x=1x = 1, f(1)=4f(1) = 4, so our **first point** is (1,4)(1, 4).

STEP 4

Next, when x=4x = 4, we see f(4)=5f(4) = 5, giving us our **second point** (4,5)(4, 5).
Awesome! We've got our two points!

STEP 5

Now, remember the **average rate of change formula**: Average rate of change=change in f(x)change in x=f(x2)f(x1)x2x1 \text{Average rate of change} = \frac{\text{change in } f(x)}{\text{change in } x} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} This formula tells us how much f(x)f(x) changes, on average, for every unit change in xx.

STEP 6

Let's **plug in** our values.
We'll call (1,4)(1, 4) our (x1,f(x1))(x_1, f(x_1)) and (4,5)(4, 5) our (x2,f(x2))(x_2, f(x_2)).
So, x1=1x_1 = 1, f(x1)=4f(x_1) = 4, x2=4x_2 = 4, and f(x2)=5f(x_2) = 5.

STEP 7

Substituting these values into the formula, we get: f(4)f(1)41=5441 \frac{f(4) - f(1)}{4 - 1} = \frac{5 - 4}{4 - 1}

STEP 8

Now, we **simplify**: 5441=13 \frac{5 - 4}{4 - 1} = \frac{1}{3} So, the **average rate of change** of ff from x=1x = 1 to x=4x = 4 is 13\frac{1}{3}!

STEP 9

The average rate of change of ff from x=1x = 1 to x=4x = 4 is 13\frac{1}{3}.

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