Calculus Examples

$f (x) = a^{x}$ , $(2, 64)$

Step 1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${a | a \in ℝ}$

Step 2

$f (x)$ is continuous on $[2, 64]$ .

$f (x)$ is continuous

Step 3

The average value of function $f$ over the interval $[a, b]$ is defined as $A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$ .

$A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$

Step 4

Substitute the actual values into the formula for the average value of a function.

$A (x) = \frac{1}{64 - 2} (\int_{2}^{64} a^{x} d x)$

Step 5

Subtract $2$ from $64$ .

$A (x) = \frac{1}{62} \cdot \int_{2}^{64} a^{x} d x$

Step 6

Combine $\frac{1}{62}$ and $\int_{2}^{64} a^{x} d x$ .

$A (x) = \frac{\int_{2}^{64} a^{x} d x}{62}$

Step 7