https://file.aiquickdraw.com/mathbot/file/1724183248349mh5fuw5p.jpg

Consider the exponential function shown here. Point A
is at (−3;−5)
and Point B
is at (−1;52)
, as shown.

What is the average gradient of g
between the points?

Answer:
The average gradient is

To find the average gradient (or average rate of change) of the function \( g(x) \) between the points \( A(-3, -5) \) and \( B(-1, 52) \), we use the following formula for the average gradient between two points \((x_1, y_1)\) and \((x_2, y_2)\) on a function:

\[
\text{Average Gradient} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the coordinates of points \( A \) and \( B \):

\[
\text{Average Gradient} = \frac{52 - (-5)}{-1 - (-3)}
\]

Now, simplify the expression:

\[
\text{Average Gradient} = \frac{52 + 5}{-1 + 3} = \frac{57}{2} = 28.5
\]

So, the average gradient of \( g(x) \) between the points \( A(-3, -5) \) and \( B(-1, 52) \) is \( 28.5 \).

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. What is the gradient of the function \( g(x) \) at a specific point, such as \( x = -2 \)?
2. How would you find the equation of the tangent to the curve \( g(x) \) at point \( B \)?
3. What is the significance of the gradient in the context of an exponential function?
4. How does the gradient change as \( x \) increases for an exponential function?
5. How do you calculate the instantaneous rate of change for \( g(x) \) at a particular point?

**Tip:** When calculating the average gradient, ensure that you correctly subtract the coordinates in the right order to avoid sign errors.

Learn how to calculate the average gradient of the exponential function g(x) between two points A(-3, -5) and B(-1, 52). Understand the concept of rate of change in exponential functions and apply the average gradient formula step-by-step.

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