Interactive-12Adv-Rates of change graphically

When the first derivative of a function, f'(x), is less than 0, what does this indicate about the graph of y = f(x)?

• It is increasing

• It is decreasing

• It has a stationary point

• It is concave upwards

If the second derivative of a function, f''(x), is greater than 0, what does this imply about the graph of y = f'(x) and the concavity of y = f(x)?

• f'(x) is decreasing and f(x) is concave downwards

• f'(x) is increasing and f(x) is concave upwards

• f'(x) is stationary and f(x) has a point of inflection

• f'(x) is decreasing and f(x) is concave upwards

For a curve that is generally moving upwards from left to right, and its slope is becoming steeper, what are the signs of its first and second derivatives?

• First derivative is positive, second derivative is positive

• First derivative is positive, second derivative is negative

• First derivative is negative, second derivative is positive

• First derivative is negative, second derivative is negative

What does it mean if a function has a positive gradient and its rate of increase is slowing?

• Concave upwards

• Concave downwards

• Stationary point

• Decreasing

If the first derivative of a population P with respect to time t (dP/dt) is less than 0, what can be concluded about the population?

• The population is increasing.

• The population is decreasing.

• The population is stable.

• The population is at a maximum.

If the second derivative of a population P with respect to time t (d²P/dt²) is greater than 0, what does this indicate about the population's rate of change?

• The rate of change is decreasing.

• The rate of change is increasing.

• The rate of change is constant.

• The population is at a minimum.

A population is decreasing, and its rate of change is increasing. Which of the following best describes the graph of this population over time?

• The graph is increasing and concave upwards.

• The graph is decreasing and concave downwards.

• The graph is decreasing and concave upwards.

• The graph is increasing and concave downwards.