Calculus Review - Applications of Derivatives

Front

What is a point of inflection?

Back

A point of inflection is where a function goes from concave up to concave down.

Front

Where does the function f(x) = 2x^3 - 9x^2 + 12x - 3 have relative max/min values?

Back

Relative max at x = 1, Relative min at x = 2.

Front

If a function has a second derivative that is positive, what does that tell you?

Back

The function is concave up.

Front

If the position function for a particle is s(t) = -t^2 - t, what is the instantaneous velocity function for the particle?

Back

v(t) = -2t - 1.

Front

Given a function g(x), if g'(x)=0 at a certain value of x, then g(x) has _____________ at x.

Back

a critical point.

Front

What does the first derivative of a function represent?

Back

The first derivative represents the instantaneous rate of change of the function, or the slope of the tangent line.

Front

What is the significance of a critical point in a function?

Back

A critical point is where the function's derivative is zero or undefined, indicating potential relative maxima, minima, or points of inflection.

Front

How do you find the relative extrema of a function?

Back

To find relative extrema, set the first derivative equal to zero and solve for x, then use the second derivative test to classify the extrema.

Front

What does it mean if the second derivative of a function is negative?

Back

If the second derivative is negative, the function is concave down.

Front

What is the relationship between the first and second derivatives in determining concavity?

Back

The first derivative indicates increasing or decreasing behavior, while the second derivative indicates concavity (up or down).