Calculus and Physics in Speed Analysis

The rate of change of a function, calculated by dividing the change in the function's output by the change in its input.

Finding the instantaneous speed by calculating the limit of the average speed as the time interval approaches zero.

The formula to find the instantaneous speed, often expressed as a derivative of position with respect to time, denoted as 'v' or 'dx/dt'.

The world record for the 100m men's race, accomplished by Usain Bolt in 9.58 seconds.

The highest speed attained by a runner during the 100m race.

The distance traveled by the runner divided by the time taken to travel that distance.

The distance traveled by a runner over a specific time interval, calculated by dividing the distance by the time interval.

A rule used to differentiate a fraction by finding the difference of the numerator and denominator, each differentiated separately, then divided by the square of the denominator.

If the first derivative of a function is positive for all x values within a specific interval, then the function is strictly increasing within that interval.

If the first derivative of a function is negative for all x values within a specific interval, then the function is strictly decreasing within that interval.

A point on a function's graph where the function reaches a maximum value compared to its neighboring points.

A point on a function's graph where the function reaches a minimum value compared to its neighboring points.

The derivative of the sine function is the cosine function.

The derivative of the cosine function is the negative sine function.

The derivative of the tangent function is the secant squared function.

The derivative of the exponential function with base 'e' is itself.

The derivative of a natural logarithm is 1 divided by the argument of the logarithm.

The derivative of a logarithmic function with base 'a' is 1 over the natural logarithm of 'a' multiplied by the argument of the logarithm.

The instantaneous rate of change of a function at a specific point.

A mathematical technique used to determine the rate of change of a function.

A point on a graph where the function changes from increasing to decreasing or vice versa. It's the highest or lowest point in a local region of the graph.

The range of x-values for which a function is increasing or decreasing.

The process of determining whether a function is increasing or decreasing within a specific interval.

A specific method to find where a function is increasing or decreasing by calculating the derivative and analyzing its sign.

A point on a function's graph where the function's slope is zero, indicating a potential maximum or minimum value.

A point on a function's graph where the function reaches a maximum value compared to its neighboring points.

A point on a function's graph where the function reaches a minimum value compared to its neighboring points.

A mathematical test to determine whether a stationary point is a maximum, minimum, or neither, using the second derivative of the function.

The rate of change of a function's slope, indicating the curve's concavity.

The value of the second derivative at a stationary point determines whether the point is a maximum, minimum, or neither.

The second derivative of a function is negative at a maximum point and positive at a minimum point.

The second derivative test analyzes the concavity of a function at a stationary point to determine whether it is a maximum, minimum, or neither.

The derivative of a function, f'(x), helps determine the function's increasing or decreasing behavior. A positive derivative indicates an increasing function, while a negative derivative indicates a decreasing function.

Critical points are the points on a function's graph where the derivative is either zero or undefined. These points are potential locations for local maxima or minima.

To find local maxima and minima, we first need to find critical points by setting the first derivative equal to zero and solving for x. We then analyze the function's behavior around each critical point to determine if it's a maximum or minimum.

To find local extrema of a function, one common approach involves finding the critical points by setting the first derivative equal to zero or undefined, and then analyzing the behavior of the function around those critical points using techniques like the first derivative test or second derivative test.