How to estimate the distance traveled from a velocity-time graph using Riemann Sums?
1. Divide the time interval into subintervals. 2. Construct rectangles with widths equal to the subinterval lengths. 3. Determine the height of each rectangle (usually the function value at the left, right, or midpoint of the subinterval). 4. Calculate the area of each rectangle. 5. Sum the areas of all rectangles to approximate the total distance.
How to find the total distance traveled given a piecewise function representing velocity?
1. Graph the piecewise function. 2. Divide the graph into geometric shapes (rectangles, triangles, trapezoids). 3. Calculate the area of each shape. 4. Sum the areas, considering signed areas, to find the total distance.
How to solve for time given accumulation and rate of change?
Use the formula $t = \frac{a}{r}$, where $t$ is time, $a$ is accumulation, and $r$ is the rate of change.
Explain how the area under a velocity-time graph relates to distance.
The area under a velocity-time graph represents the total distance traveled.
What does a definite integral calculate?
The exact area under a curve between two specified limits.
Why are Riemann Sums approximations?
Riemann Sums use rectangles to approximate the area, which may not perfectly match the curve, leading to over or underestimations.
How do integrals 'undo' derivatives?
Integration is the reverse process of differentiation; it finds the original function before differentiation.
Define accumulation of change.
The sum, over time, of how much something has changed.
What is a Riemann Sum?
A method of approximating the area under a curve by dividing it into rectangles or other simple shapes.
Define definite integral.
An integral with upper and lower limits, representing the area under a curve between those limits.
What are integrals also called?
Anti-derivatives.
What is the meaning of 'signed area'?
Area below the x-axis is considered negative, representing a decrease or movement in the opposite direction.
