Calculus Concepts: Increasing/Decreasing, Derivatives, Theorems

These flashcards cover key concepts in calculus regarding increasing/decreasing functions, the first and second derivative tests, and implicit differentiation.

When is a function increasing?

When f′(x) > 0.

When is a function decreasing?

When f′(x) < 0.

What do you analyze to determine increasing/decreasing intervals?

The sign of f′(x).

What does the First Derivative Test classify?

Local maxima and minima by checking sign changes in f′.

What sign change indicates a local maximum?

f′ changes from + to –.

What sign change indicates a local minimum?

f′ changes from – to +.

What if f′ doesn’t change sign?

No local extremum at that point.

What does f″(x) > 0 tell you?

The function is concave up.

What does f″(x) < 0 tell you?

The function is concave down.

Where do inflection points occur?

Where concavity changes sign (f″ changes sign).

What conditions must hold to use the second derivative test?

f′(c) = 0 and f″(c) exists.

What does f″(c) > 0 mean?

Local minimum (concave up).

What does f″(c) < 0 mean?

Local maximum (concave down).

What if f″(c) = 0?

Inconclusive → use the First Derivative Test.

What does Rolle’s Theorem guarantee?

There exists a c in (a, b) where f′(c) = 0.

What conditions must be met for Rolle’s Theorem?

Function is continuous on [a, b], differentiable on (a, b), and f(a) = f(b).

What does the Mean Value Theorem guarantee?

There exists a c in (a, b) where f′(c)=f(b)−f(a)/(b−a).

What does MVT equate?

Instantaneous rate of change = average rate of change.

What conditions must be met for MVT?

Function is continuous on [a, b] and differentiable on (a, b).

What technique is used to find slope for an implicit equation?

Implicit differentiation.

What indicates a horizontal tangent in an implicit curve?

dy/dx = 0 (numerator = 0).

What indicates a vertical tangent?

dy/dx undefined (denominator = 0).

When differentiating y² with respect to x, what is the derivative?

2y (dy/dx).

What kind of behaviors can implicit curves have that normal functions cannot?

Loops, multiple y-values for one x, cusps, vertical tangents.