Contextual Applications of Differentiation

At an inflection point, the slope of the function's tangent line is zero.

At an inflection point, the function reaches its maximum or minimum value.

At an inflection point, the function crosses its horizontal asymptote.

At an inflection point, the concavity of the function changes.

Profits are increasing faster after 10 years than they were after 5 years.

Profits started to decrease after year 10.

There was no profit in year five and ten.

Total profits are higher after 10 years than after 5 years.

Function g(t) must pass through point (t,r).

The second derivative of function g(t), denoted as $g''(t)$, must also equal r.

The rate of change or slope of g(t) at any given time t corresponds to r.

Function g(t)'s graph forms an arc with radius r centered at t.

The maximum height reached by the balloon up to time $x = 3$.

The acceleration of the balloon at time $x = 3$.

The rate at which the height of the balloon is changing at the moment when $x = 3$.

The height of the balloon above the ground when $x = 3$.

The instantaneous rate of change of speed at time $t$.

The average speed of the particle from time $0$ to time $t$.

The total distance traveled by the particle from time $0$ to time $t$.

The average acceleration of the particle over time interval $[t, t+h]$.

The rate of change of the function at a specific point

The maximum or minimum points of the function

The value of the function at a specific point

The integral of the function over a specific interval

Second Derivative Test

Intermediate Value Theorem

Mean Value Theorem

First Derivative Test

The maximum point

The minimum point

The critical point

The definite integral

Nothing conclusive without more information about behavior around this point can be determined from this single fact alone.

The slope of f will remain constant everywhere since its derivative equals k at one point.

Function f has an inflection point where its first derivative equals k.

Function f must be linear because its derivative at one point is constant.

Increase both volume, height.

Volume increases while height diminishes.

Water volume decreases as height decreases.

Water volume stays stable regardless of height magnitude.