Explain the concept of accumulation in the context of integrals.
Accumulation refers to the process of summing up the values of a function over an interval to find the total change or amount.
Explain why the integral of a rate of change function gives the net change.
The integral sums up the infinitesimal changes in the quantity, resulting in the overall change between the initial and final points.
Explain how initial conditions are used in accumulation problems.
Initial conditions provide a starting point, allowing us to determine the absolute amount at any time by adding the accumulated change to the initial value.
Describe the relationship between the area under a curve and the definite integral.
The definite integral calculates the signed area between the curve and the x-axis over a given interval. Areas above the x-axis are positive, and areas below are negative.
Explain why understanding units is crucial in accumulation problems.
Units help ensure that the integral is set up and interpreted correctly, and that the final answer has the appropriate units.
Explain the meaning of a negative value for a definite integral.
A negative value indicates that the function is predominantly below the x-axis or that the quantity is decreasing over the interval.
Explain the significance of the Fundamental Theorem of Calculus in accumulation problems.
The Fundamental Theorem of Calculus connects differentiation and integration, allowing us to evaluate definite integrals using antiderivatives.
Explain how to determine the total distance traveled, given a velocity function.
Integrate the absolute value of the velocity function over the given interval: $\int_{a}^{b} |v(t)| dt$.
Explain the difference between displacement and total distance traveled.
Displacement is the net change in position, while total distance traveled considers the absolute value of the velocity, accounting for changes in direction.
Explain how to use a calculator to evaluate a definite integral.
Use the numerical integration function (fnInt) on the calculator, specifying the function, variable, and limits of integration.
What is an accumulation function?
A function that represents the accumulated amount of a quantity as its rate of change is integrated over time.
What does the definite integral represent graphically?
The area under the curve of a function between two specified limits.
What is the rate of change function?
A function that describes how a quantity changes with respect to another variable, often time.
What is displacement?
The net change in position of an object.
What does the integral of velocity represent?
Displacement (or net change in position).
What is an initial condition?
The value of a function at a specific point, often at time t=0, used to determine the constant of integration.
Define net change.
The difference between the final and initial values of a quantity.
What are the units of a rate of change function?
Units of the dependent variable per unit of the independent variable (e.g., mosquitoes per day).
What is the relationship between position, velocity, and acceleration?
Velocity is the derivative of position, and acceleration is the derivative of velocity.
What is the purpose of evaluating a definite integral?
To find the net change or accumulation of a quantity over a specific interval.
What is the formula for displacement given a velocity function $v(t)$ from $t=a$ to $t=b$?
$\int_{a}^{b} v(t) dt$
What is the formula for finding the total amount at time $t$ given a rate of change $R(t)$ and initial amount $A(0)$?
$A(t) = A(0) + \int_{0}^{t} R(x) dx$
What is the general form of a definite integral?
$\int_{a}^{b} f(x) dx$
What is the power rule for integration?
$\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$
What is the formula for the net change of a function $F(x)$ from $x=a$ to $x=b$?
$F(b) - F(a) = \int_{a}^{b} F'(x) dx$
What is the integral of $\cos(x)$?
$\int \cos(x) dx = \sin(x) + C$
What is the integral of $\sin(x)$?
$\int \sin(x) dx = -\cos(x) + C$
What is the formula to find the number of mosquitoes at t=31?
$1000 + \int_{0}^{31} R(t) dt$
What is the integral of a constant $k$ with respect to $x$?
$\int k dx = kx + C$
What is the formula for integrating a function multiplied by a constant?
$\int kf(x) dx = k \int f(x) dx$
