CPM Calc Ch. 5 Review

Front

What is the derivative of a function?

Back

The derivative of a function measures how the function's output value changes as its input value changes. It represents the slope of the tangent line to the function's graph at a given point.

Front

What does f'(c) = 0 indicate about the function f at x = c?

Back

It indicates that there is a critical point at x = c, which could be a local maximum, local minimum, or a saddle point.

Front

What does f''(c) < 0 imply about the function f at x = c?

Back

It implies that the function is concave down at that point, suggesting that there is a local maximum at x = c.

Front

What is linearization in calculus?

Back

Linearization is the process of approximating a function near a point using the tangent line at that point.

Front

How do you find the linearization of f(x) at x = a?

Back

The linearization L(x) is given by L(x) = f(a) + f'(a)(x - a).

Front

What is the product rule in differentiation?

Back

The product rule states that if you have two functions u(x) and v(x), then the derivative of their product is u'v + uv'.

Front

What is the chain rule in differentiation?

Back

The chain rule states that if you have a composite function f(g(x)), then the derivative is f'(g(x)) * g'(x).

Front

What is a local maximum?

Back

A local maximum is a point where the function value is higher than the values of the function at nearby points.

Front

What is a local minimum?

Back

A local minimum is a point where the function value is lower than the values of the function at nearby points.

Front

What does it mean if f''(c) > 0?

Back

It means that the function is concave up at x = c, indicating a local minimum at that point.