CALC: finding critical points Flashcards | Quizlet

How do you find critical numbers?

Find the first derivative of the function, set it equal to zero, and solve for $x$.

How do you find critical points after finding the critical numbers?

Plug the critical numbers into the ORIGINAL function to find the $y$-value. The critical number ($x$-value) and the value found ($y$-value) are the coordinates for the critical point.

In the example where a critical number is $1$ and the original function is $f(x) = x^2 + 4$, how do you find the critical point?

Plug in $1$ for $x$ into the original function to calculate the $y$-value, pairing it with $1$ as the coordinates.

What is the chain rule procedure described in the notes?

Bring down power, multiply by whatever is left, and multiply by the derivative of whatever is left.

How are roots identified on a graph based on their power?

Single root when only variable, double root when squared, and triple root when cubed.

What is the derivative of $\sin(x)$?

$$\cos(x)$$

What is the derivative of $\cos(x)$?

$$-\sin(x)$$

What are absolute extrema?

Absolute minimum and absolute maximum.

What are the steps to find the absolute extrema of a function on a given interval (e.g., $[-1, 2]$)?

Find the critical numbers ($f'(x) = 0$), plug the interval endpoints and critical numbers into the original function, and evaluate the resulting $y$-values: the lowest is the absolute min and the greatest is the absolute max.

When using the first derivative line test, what does it mean if all the values between critical numbers have the same sign?

There are no relative max or min.

On a first derivative line test, what sign change indicates a relative minimum?

The values go from negative to positive.

On a first derivative line test, what sign change indicates a relative maximum?

The values go from positive to negative.

How do you find the $y$-value of a relative maximum or relative minimum coordinate?

Plug the $x$-value where the extrema takes place into the ORIGINAL function.

Concave up

Slope is increasing.

Concave down

Slope is decreasing.

What is the test for concavity using the second derivative?

If $f''(x) > 0$, then $f$ is concave up; if $f''(x) < 0$, then $f$ is concave down.

What does an interval inequality like $-84x^2 < 0$ indicate about a function's concavity?

It shows that at this interval the function is concave down.

What is a point of inflection?

A point where the graph of a function has a tangent line and where the concavity changes.

How do you find possible points of inflection for a function?

Find the second derivative of the function and set it equal to $0$.

How do you determine the intervals of concavity using possible points of inflection?

Plot the possible points of inflection on a number line, evaluate the second derivative for numbers in between, and classify positive values as concave up and negative values as concave down.

How do you find the coordinates for the final points of inflection?

Plug the confirmed $x$-value of the inflection point into the original equation to find the corresponding $y$-value.

In the second derivative test, what does it mean if the second derivative evaluated at a critical number is greater than zero ($f''(x) > 0$)?

There is a relative minimum at that point.

In the second derivative test, what occurs if $f''(x) < 0$ and what occurs if $f''(x) = 0$?

If $f''(x) < 0$, there is a relative max at that point; if $f''(x) = 0$, the test fails (it does not count as a relative min or max).

What does it mean graphically when the derivative of a function does not exist at a point?

There is a sharp turn on the graph.

How do you find the intervals where a function is increasing and decreasing?

By using the first derivative number line test (when finding relative max and min).