Calculus Final Exam Review (semester 1)

Front

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

Back

A function f is concave up on an interval if its second derivative f''(x) is positive on that interval. This indicates that the graph of f is curving upwards.

Front

How can you determine the intervals where a function is concave up using its first derivative f'?

Back

A function f is concave up where its first derivative f' is increasing. This can be determined by analyzing the graph of f'.

Front

What is a point of inflection?

Back

A point of inflection is a point on the graph of a function where the concavity changes, which occurs when the second derivative f'' changes sign.

Front

How do you find points of inflection using the first derivative f'?

Back

Points of inflection can be found by identifying where the first derivative f' changes from increasing to decreasing or vice versa.

Front

What is the relationship between the first derivative f' and the second derivative f''?

Back

The first derivative f' indicates the slope of the function f, while the second derivative f'' indicates the concavity of the function.

Front

If f' is positive, what can be said about the function f?

Back

If f' is positive, the function f is increasing on that interval.

Front

If f' is negative, what can be said about the function f?

Back

If f' is negative, the function f is decreasing on that interval.

Front

What does it mean for f' to be increasing?

Back

If f' is increasing, it means that the slope of the function f is becoming steeper, indicating that the second derivative f'' is positive.

Front

What does it mean for f' to be decreasing?

Back

If f' is decreasing, it means that the slope of the function f is becoming less steep, indicating that the second derivative f'' is negative.

Front

How do you determine the intervals of increase and decrease for a function f?

Back

To determine intervals of increase and decrease, analyze the sign of the first derivative f'. If f' > 0, f is increasing; if f' < 0, f is decreasing.