Contextual Applications of Differentiation

The tangent line and the function have the same value but different slopes at the point of tangency

The tangent line and the function have neither the same value nor the same slope at the point of tangency

The tangent line and the function have the same slope but different values at the point of tangency

The tangent line and the function have the same value and slope at the point of tangency

The accuracy becomes undefined when the function becomes more nonlinear

The accuracy remains the same regardless of the function's nonlinearity

The accuracy decreases as the function becomes more nonlinear

The accuracy increases as the function becomes more nonlinear

Actual Calculation because precision deteriorates rapidly outside the immediate vicinity of the convergence zone imposed by limit conditions, thereby skewing outcomes unfavorably compared to differential approaches

Differential estimation due to its reliance on known slopes rather than possibly fluctuating rates of change

Actual Calculation owing to the inherent stability embedded in the process, especially in the context of singularities and discontinuities that may arise unexpectedly, throwing off the delicate balance required for effective differential analysis

Differential Estimation since the inherently uncertain nature of derivatives surrounding limits makes interpretive assessments riskier than straightforward arithmetic resolutions offered by actual calculation procedures

It becomes more accurate as h approaches zero.

The slope used in approximation can be any number close to f(c).

Its accuracy depends on higher derivatives like $f''(c)$.

It gives exact values regardless of how far from c you evaluate it.

To find the exact value of a function at a given point

To approximate a complicated function using a simpler linear function

To determine the concavity of a function

To calculate the integral of a function over a specific domain

The instantaneous rate of change of the function at that point.

The maximum value of the function near that point.

The minimum value of the function near that point.

The average rate of change of the function over an interval.

$7-$

$9-$

$8.80$

$9.20$

To calculate the definite integral of functions

To determine the exact values of functions at specific points

To analyze the concavity and inflection points of functions

To approximate and estimate values that are difficult or impossible to calculate directly

There must be a discontinuity in $g(x)$ at $x$ equals $c$.

The value of $g(c)$ must equal $L$ when $x$ equals $c$.

The values of $g(x)$ get arbitrarily close to $L$ as $x$ gets closer to $c$.

The slope of $g(x)$ approaches zero as $x$ approaches $c$.

Tangent line at $x=c$.

Secant line through points around $x=c$.

Vertical axis passing through $x=c$.

Horizontal asymptote near $x=c$.