Fundamentals of Differentiation

$\int_{}{} F'(t) , dt$

$\int_{a}^{b} F(t) , dt$

$\sum_{}{} F(t) \Delta t$

F(b) - F(a)

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 $a$.

The function should have a maximum or minimum at the point of interest

The function should have a continuous graph at the point of interest

The function should be defined at the point of interest

The function should be differentiable at the point of interest

The technology method

The difference quotient method

The average rate of change method

The tangent line method

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A tangent line slope calculated as $\frac{q(4 + h) - q(4)}{h}$ with $h$ approaching zero.

The net change from $p=3$ to $p=5$ divided by total time elapsed $\int_{3}^{5} q(p)\ dp$.

An average rate over interval $[p-4,p+4]$ represented by $\frac{q(p+4)-q(p-4)}{8}$.

$q'(p)\cdot dp|_{p=4}$.

The antiderivative of a function

The rate of change of a function at a specific point

The limit of a function

The area under a curve

By finding the average rate of change over an interval

By drawing a tangent line to the graph at a specific point

By calculating the difference quotient

By inputting the function and specifying the point of interest

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