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The instantaneous slope of a curve is the slope of that curve at a single point. In calculus, this is called the derivative. It also might be called the line tangent to the curve at a point.
If you imagine an arbitrary curve (just any curve) with two points on it (point P and point Q), the slope between P and Q is the slope of the line connecting those two points. This is called a secant line. If you keep P where it is and slide Q closer and closer to P along the curve, the secant line will change slope as it gets smaller and smaller. When Q gets extremely close to P (so that there is an infinitesimal space between P and Q), then the slope of the secant line approximates the slope at P. When we take the limit of that tiny distance as it approaches zero (meaning we make the space disappear) we get the slope of the curve at P. This is the instantaneous slope or the derivative of the curve at P.
Mathematically, we say that the slope at P = limh→0 [f(x+h) - f(x)]÷h = df/dx, where h is the distance between P and Q, f(x) is the position of P, f(x+h) is the position of Q, and df/dx is the derivative of the curve with respect to x.
The formula above is a specific case where the derivative is in terms of x and we're dealing with two dimensions. In physics, the instantaneous slope (derivative) of a position function is velocity, the derivative of velocity is acceleration, and the derivative of acceleration is jerk.
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What is Instantaneous rate of change?
It is the rate of change at one given moment, and it is the same as the value of the derivative at a particular point. The point may be thought of as that given moment. When we talk about functions, the instantaneous rate of change at a point is the same as the slope, m, of the tangent line.. Sometimes we think of it as the slope of the curve. The best way to understand this is with the difference quotient and limits. The difference quotient is the average rate of change of y with respect to x. If we then look at the difference quotient and we let delta x ->0, this will be the instantaneous rate of change. In other words, the time interval gets smaller and smaller. Difference quotient is delta y/ delta x where delta represents the change.
What is a derivative in calculus?
When you take the derivative of a function, you are seeking a variation of that function that provides you with the slope of the tangent (instantaneous slope) at any value of (x). For example, the derivative of the function f(x)=x^2 is f'(x)=2x. Notice that the derivative is denoted by the apostrophe inside the f and (x). Also note that at x=0, f'(x)=0, which means that at x=0 the slope of the tangent is zero, which is correct for the function y=x^2.
Y equals 3x plus 5 slope?
The slope equals 3
What is the definition of a point-slope form?
point slope form is y-y1=m(x-x1). x1 and y1 are both points and m is the slope.
What is The Slope Of -5x plus 3y equals plus 3?
Convert the equation to the standard slope intercept form of y = mx + c, where the value of m is the slope. -5x + 3y = 3 3y = 5x + 3 y = 5/3x + 1 Then the slope is 5/3.
What does the slope of the instantaneous speed-vs-time graph represented?
The slope of the instantaneous speed-vs-time graph represents the acceleration of the object. A positive slope indicates the object is accelerating in the positive direction, while a negative slope indicates acceleration in the negative direction. The steeper the slope, the greater the magnitude of the acceleration.
Does the slope of a position time graph represent average velocity or rate of change of velocity?
The slope of a position-time graph represents the average velocity of an object. It does not represent the rate of change of velocity, which would be represented by the slope of a velocity-time graph.
What is the difference between a slope and rate of change?
Depends. Slope of tangent = instantaneous rate of change. Slope of secant = average rate of change.
What is the slope of a staircase?
Average or instantaneous? The average is the height divided by the horizontal distance, whereas the instantaneous alternates between zero and infinity.
Slope of a displacement time graph?
It is the instantaneous speed in the direction in which the displacement is measured.
What does the slope of a distance vs time graph indicate about an object's motion?
instantaneous velocity
What does the slope of distance-versus-time graph represent physically?
instantaneous magnitude of velocity
How is instantaneous acceleration related to a velocity-time graph?
Instantaneous acceleration at any point on a velocity-time graph can be determined by calculating the slope of the tangent line at that specific point. A steeper slope represents a higher acceleration, while a shallower slope indicates a lower acceleration.
Short-lived velocity defined as the slope of the tangent line at a given point on a curve?
instantaneous
What is the instantaneous rate of the reaction at t equals 800 seconds?
The instantaneous rate of a reaction at t=800 seconds can be determined by calculating the slope of the tangent line to the concentration-time curve at that specific point in time. This slope represents the rate of the reaction at that moment, giving you the instantaneous rate at t=800 seconds.
The slope of a line drawn tangent to a point on the curve of a position vs time graph describes what concept?
The slope of a line drawn tangent to a point on a position vs. time graph represents the instantaneous velocity of the object at that point. It describes how the position of the object is changing at that exact moment in time.
How can the rate law for a a chemical reaction be used to determine an instantaneous reaction rate?
The rate law for a chemical reaction expresses how the rate of the reaction depends on the concentration of reactants. By plugging in the instantaneous concentrations of the reactants into the rate law equation, we can calculate the instantaneous reaction rate at a specific moment in time.
