Graphs of functions and their derivatives

Front

What does a negative derivative indicate about a function?

Back

The function is decreasing.

Front

What is the derivative of a function?

Back

The instantaneous rate of change of the function.

Front

What is the derivative of the function f(x) = 5x^2?

Back

f'(x) = 10x.

Front

If a function's derivative is positive, what can we say about the function?

Back

The function is increasing.

Front

What does it mean if the derivative of a function is zero at a point?

Back

The function has a local maximum, minimum, or a point of inflection at that point.

Front

What is the relationship between a function and its derivative?

Back

The derivative provides information about the function's rate of change and slope.

Front

How do you find the derivative of a polynomial function?

Back

Use the power rule: if f(x) = ax^n, then f'(x) = nax^(n-1).

Front

What is the power rule for differentiation?

Back

If f(x) = x^n, then f'(x) = nx^(n-1).

Front

What is the significance of the second derivative?

Back

The second derivative indicates the concavity of the function and can identify points of inflection.

Front

What does it mean for a function to be concave up?

Back

The second derivative is positive, indicating that the function is curving upwards.