As an exercise, suppose that I tell you that the instantaneous rate of change of the function #f(x) = x^3 -1# at the point #x=2# is 12. You can estimate this by using intervals of different size. In this case, try first the interval #[-1,5]# followed by the interval #[0,4]# and then the interval #[1,3]#. You should notice that as your interval ...

The instantaneous rate of change, or derivative, can be written as dy/dx, and it is a function that tells you the instantaneous rate of change at any point. #y' = f'(x+h) = (d/(dx)) (3*(x)^2) = 6x * 1 = 6x#. For example, if x = 1, then the instantaneous rate of change is 6.

2017年12月2日 · You approximate it by using the slope of the secant line through the two closest values to your target value. Sometimes there are more requirements, but in AP Calculus you almost always end up taking the values to the left and right of your target value and calculating based on them. x | f(x) 1 | 3 2 | 5 3 | 4 4 | 9 From the "table" we'd approximate the following: …

2014年7月31日 · Instantaneous rate of change of a function is represented by the slope of the line, it tells you by how much the function is increasing or decreasing as the #x#-values change. Figure 1. Slope of a line. In this image, you can see how the blue function can have its instantaneous rate of change represented by a red line tangent to the curve.

2014年8月4日 · The instantaneous rate of change at a point is equal to the function's derivative evaluated at that point. In other words, it is equal to the slope of the line tangent to the curve at that point. For example, let's say we have a function f(x) = x^2. If we want to know the instantaneous rate of change at the point (2, 4), then we first find the derivative: f'(x) = 2x And …

2014年8月12日 · Yes, it is possible for the instantaneous rate of change to be 0. For a specific example, imagine the function f(x) = 3. This is a horizontal line parallel to the x-axis at the value y=3. This function is unchanging for any value of x, therefore its rate of change is zero. For a physics example, if I stand still in a spot for 5 minutes, then my rate of change of position after …

2014年9月21日 · The instantaneous rate of change of a function f(x) at a point x=a is f'(a), which is the derivative of the function f evaluated at x=a.

2014年8月12日 · Most certainly! When the instantaneous rate of change of a function at a given point is negative, it simply means that the function is decreasing at that point. As an example, given a function of the form y=mx+b, when m is positive, the function is increasing, but when m is negative, the function is decreasing. For a line, the rate of change at any given point is simply …

2014年8月12日 · As an exercise, suppose that I tell you that the instantaneous rate of change of the function #f(x) = x^3 -1# at the point #x=2# is 12. You can estimate this by using intervals of different size. In this case, try first the interval #[-1,5]# followed by the interval #[0,4]# and then the interval #[1,3]#. You should notice that as your interval ...

2017年9月15日 · The instantaneous rate of reaction is the slope of the line (the tangent to the curve) at any time t. (From SPM Chemistry) How do we determine it? We draw the best tangent to the line that we can and extend it to convenient points on the axis that will make it easy to calculate the value of (Δ["P"])/(Δt) For example, the graph below shows the ...