All Flashcards
Explain how the Fundamental Theorem of Calculus connects area and antiderivatives.
The area under a function's curve is equal to the value of its antiderivative, calculated with the same bounds.
How do you determine where a function is increasing using its derivative?
A function is increasing where its first derivative is positive.
How do you determine where a function is decreasing using its derivative?
A function is decreasing where its first derivative is negative.
How do you find relative extrema using the first derivative?
Relative extrema occur where the first derivative is zero or undefined and changes sign.
How do you determine concavity using the second derivative?
A function is concave up where its second derivative is positive and concave down where it is negative.
How do you find inflection points using the second derivative?
Inflection points occur where the second derivative changes sign.
What does the area below the x-axis represent when calculating definite integrals?
Area below the x-axis is considered negative when calculating definite integrals.
Explain how the graph of relates to the increasing/decreasing behavior of .
When , is increasing. When , is decreasing.
Explain how the graph of relates to the concavity of .
The slope of indicates the concavity of . Positive slope means concave up, negative slope means concave down.
Explain how to find absolute minimum/maximum on a closed interval.
Evaluate the function at critical points and endpoints; the smallest/largest value is the absolute minimum/maximum.
Given the graph of , what does the area between the curve and the x-axis represent?
The area represents the change in the original function over the given interval.
Given the graph of , how do you identify intervals where is increasing?
is increasing where is above the x-axis (positive).
Given the graph of , how do you identify intervals where is decreasing?
is decreasing where is below the x-axis (negative).
Given the graph of , how do you identify relative extrema of ?
Relative extrema occur where crosses the x-axis, changing sign.
Given the graph of , how do you identify inflection points of ?
Inflection points occur where the slope of changes sign.
What does the x-intercept of the graph of represent?
It represents a critical point of the original function .
How can you find the value of if you have the graph of and the value of ?
Use the formula . The integral is the area under the curve of from to .
Given the graph of , how do you determine the concavity of ?
If is increasing, is concave up. If is decreasing, is concave down.
Given the graph of , what does a sharp corner or cusp indicate about ?
It indicates a possible point where is undefined, but it's not necessarily an inflection point unless the concavity changes.
Given the graph of , how do you determine where has a local maximum?
Look for points where changes from positive to negative.
Define Accumulation Function.
A function that represents the accumulated area under a curve from a fixed point to a variable point.
What is an antiderivative?
A function whose derivative is the given function.
Define the Fundamental Theorem of Calculus.
The theorem that links the concept of the integral of a function with the concept of the derivative of a function.
What is a point of inflection?
A point on a curve where the concavity changes.
Define relative maximum.
A point where the function's value is greater than or equal to the values at all nearby points.
Define relative minimum.
A point where the function's value is less than or equal to the values at all nearby points.
Define Concave Up.
A curve that opens upwards.
Define Concave Down.
A curve that opens downwards.
What is the area under the curve?
The integral of a function between two points, representing the accumulation of the function's values.
Define critical point.
A point where the derivative of a function is either zero or undefined.