Calculus Derivatives and Critical Points Quiz

What is the 4th derivative of the function $f(x) = 3x$?

What is notable about the derivatives of the function $f(x) = e^x$?

For the function $f(x) = rac{1}{ ext{s}}$ (replacing $x$ with the variable in question), what can you say about the signs of the derivatives?

What happens to the linear term in the function $f(x) = x^3 + 2x$ after the second derivative?

According to the Increasing/Decreasing (I/D) Test, what can be said about a function if $f'(x) < 0$ for all $x$ in an interval?

What is the derivative of f(x) = ln x?

In which interval is the function f(x) = ln x defined?

What can be concluded about the function g(x) = x^2 for x < 0?

At what point does g(x) = x^2 have a critical point?

What does the I/D test help determine?

Derivatives Applications

The presentation covers various applications of derivatives, including higher-order derivatives, increasing/decreasing tests for monotonicity, maximum and minimum values, the second derivative test, L'Hôpital's rule, exercises, and concluding remarks.

Higher Derivatives

A differentiable function, y = f(x), has derivatives of various orders.
The first derivative, dy/dx or y' = f'(x), shows the function's rate of change.
The second derivative, d²y/dx² or y" = f"(x), represents the rate of change of the rate of change.
The nth derivative, dⁿy/dxⁿ or y(n) = f⁽ⁿ⁾(x), is the nth derivative of the function.

Increasing/Decreasing (I/D) Test for Monotonicity

A function f is increasing on an interval I if f'(x) > 0 for all x ∈ I.
A function f is decreasing on an interval I if f'(x) < 0 for all x ∈ I.

Maximum and Minimum Values

A local maximum occurs at point c if f(c) ≥ f(x) for all x near c.
A local minimum occurs at point c if f(c) ≤ f(x) for all x near c.

Second Derivative Test

If f′(c) = 0 and f″(c) > 0, then f has a local minimum at c.
If f′(c) = 0 and f″(c) < 0, then f has a local maximum at c.
This test only applies at critical points.

L'Hôpital's Rule

L'Hôpital's rule is used to evaluate limits of indeterminate form 0/0 or ∞/∞.
This rule states: If lim f(x) = 0 and lim g(x) = 0 (or both limits are ∞), then lim f(x)/g(x) = lim f'(x)/g'(x).

Exercises

The presentation includes examples and problems related to each topic discussed.

Conclusion

The presentation summarizes the key concepts covered regarding derivatives and their various applications.

Test your understanding of derivatives, critical points, and function behavior with this quiz. Questions cover topics such as higher-order derivatives and increasing/decreasing functions. Perfect for students looking to reinforce their calculus knowledge.