Fundamentals of Differentiation

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It reduces the instantaneous rate of change to one-third at $x = 2$.

It increases the instantaneous rate of change sixfold at $x = 2$.

It has no effect on the instantaneous rate of change at $x = 2$.

It triples the instantaneous rate of change at $x = 2$.

The instantaneous rate of change of $f$ at point $a$

The slope of the secant line through points on the graph near $a$

The total change in function values over an interval containing $a$

The area under the curve from point $0$ to point $a$

Line that perpendicular to tangent line.

A line that pass (x1, y1) and (x2, y2) of curve with a slope equal to average rate of change between point (x1, y1) and (x2, y2).

A line with an instantaneous slope at a point

Curve lines.

The object’s acceleration becomes $\alpha$ times greater for each value of t.

The direction in which an object is moving reverses for each value of t.

The object’s velocity remains unchanged for each value of t.

The object's velocity becomes $\alpha$ times faster for each value of t.

$3x^2 + c$

$12$

$6x$

$\frac{1}{4}x^{-2}$

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2

-5

A constant value equal to $\frac{g(b)-g(a)}{b-a}$ for all points in [a,b].

The derivative, denoted as ${g}'(t)$ at a specific t-value.

The slope between any two distinct points on g(t).

The integral from a to b, denoted as $\int_a^b g(t) ,dt$.

$\frac{15 + 6}{5 + 2}$

$\frac{15 - 6}{5 - 2}$

$\frac{5 - 2}{15 - 6}$

$\frac{f(5)}{f(2)}$

Using the derivative of $f(x)$ evaluated at $x=c$.

Calculating the average rate of change between two points close to $x=c$.

Applying the Mean Value Theorem over an interval containing $x=c$.

Estimating using a tangent line graphically drawn at the point $(c, f(c))$.