All Flashcards
Explain accumulation of change.
The net change in a function's value over an interval, found by integrating its rate of change (derivative).
Explain the relationship between integrals and derivatives.
Integration and differentiation are inverse operations; one 'undoes' the other.
Explain how to find units for accumulation problems.
Multiply the units of the y-axis by the units of the x-axis.
What does the sign of the area represent when evaluating integrals?
Area above the x-axis is positive, and area below the x-axis is negative, considering the direction of integration.
How does increasing the number of subintervals affect Riemann Sums?
Increasing the number of subintervals generally leads to a better approximation of the area under the curve.
What are inflection points?
Inflection points occur where the second derivative changes sign.
When does a Right Riemann Sum overestimate the area?
For increasing functions, a Right Riemann Sum will always lead to an overestimate.
When does a Right Riemann Sum underestimate the area?
For decreasing functions, a Right Riemann Sum will always lead to an underestimate.
What does the numerical bound have on the answer when using the Fundamental Theorem of Calculus?
The numerical bound has no impact on the answer.
What is partial fraction decomposition?
The process of undoing a common denominator so that a rational function can be re-written as the sum of rational functions with linear denominators.
How to approximate area under a curve using Riemann Sums?
Divide the interval into subintervals, create rectangles/trapezoids, calculate their areas, and sum them up.
How to evaluate a definite integral using FTC?
Find the antiderivative of the function, evaluate it at the upper and lower limits, and subtract the values.
How to find the area given a rate of change function?
Integrate the rate of change function over the given interval.
How to find the total distance traveled given a velocity function?
Integrate the absolute value of the velocity function over the given interval.
How to use u-substitution to solve an integral?
Choose u, find du, rewrite the integral in terms of u, integrate, and substitute back to the original variable.
How to find the value of C in an indefinite integral?
Use the initial condition given, substitute into the equation, and solve for C.
How to evaluate a definite integral with u-substitution?
Choose u, find du, change the bounds, rewrite the integral in terms of u, integrate, and evaluate using the new bounds.
How to solve integrals using long division?
Use long division to rewrite the integral, then integrate term by term.
How to solve integrals by completing the square?
Rewrite the denominator by completing the square, then use u-substitution and inverse trig functions.
How to solve integrals using integration by parts?
Choose u and dv, find du and v, and apply the formula: .
What are the differences between definite and indefinite integrals?
Definite: has bounds, results in a number. | Indefinite: no bounds, results in a function + C.
Compare Left, Right, and Midpoint Riemann Sums.
Left: uses left endpoint for height. | Right: uses right endpoint for height. | Midpoint: uses midpoint for height.
Compare overestimation and underestimation of Riemann Sums.
Overestimate: Area approximation is greater than the actual area. | Underestimate: Area approximation is less than the actual area.
What are the differences between relative and absolute extrema?
Relative: Local max/min within an interval. | Absolute: Overall max/min over the entire domain.
What are the differences between convergent and divergent improper integrals?
Convergent: The limit exists and is a real number. | Divergent: The limit does not exist or is infinite.