Fundamentals of Differentiation

It changes the domain of the function

It decreases the accuracy of the estimate

It has no effect on the accuracy of the estimate

It increases the accuracy of the estimate

The limit of $\left( \frac{(f(\infty)+\Delta)}{a} \right)$ as $\Delta x$ approaches infinity.

The limit of $\left( \frac{(f(a))^{\Delta x}}{a} \right)$ as $\Delta x$ approaches a.

The limit of $\left( \frac{(f(a)+\Delta x)}{a} \right)$ as $\Delta x$ approaches a small positive value.

$\frac{d}{dx}(f(a))$ evaluated at $a$.

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1

2

4

It allows us to calculate the area under a curve

It allows us to approximate the rate of change of a function at a specific point

It provides the exact derivative of a function at a poin

It helps us find the maximum and minimum values of a function

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3

0

-6

Finding the average rate of change over an interval

Drawing a tangent line to the graph at a specific point

Calculating the difference quotient

Using technology such as a calculator or software

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26

12

32

Making predictions or conclusions about the behavior of the function

Finding the exact derivative of the function

Using technology to estimate the derivative

Approximating the average rate of change over an interval

$\pi\left(\frac{x^2}{2}\right)\bigg|_0^4$

$\pi\left(\frac{x^3}{3}\right)\bigg|_0^4$

$\pi\int_{0}^{4} x^2 , dx$

$\int_{0}^{4}\pi x , dx$

The original function must have no real roots.

The second-derivative test fails here.

The function is decreasing around $x=a$.

The function has an inflection point around $x=a$.