Table of Contents
What does the derivative of a curve tell you?
The derivative tells us if the original function is increasing or decreasing. Because f′ is a function, we can take its derivative. The second derivative gives us a mathematical way to tell how the graph of a function is curved. The second derivative tells us if the original function is concave up or down.
What second derivative tells us?
By taking the derivative of the derivative of a function f, we arrive at the second derivative, f′′. f ″ . The second derivative measures the instantaneous rate of change of the first derivative. The sign of the second derivative tells us whether the slope of the tangent line to f is increasing or decreasing.
What’s the formula for area under the curve?
The area under a curve between two points is found out by doing a definite integral between the two points. To find the area under the curve y = f(x) between x = a & x = b, integrate y = f(x) between the limits of a and b. This area can be calculated using integration with given limits.
What is the area under the curve of a function?
Suppose you have a function that graphs velocity on the y axis and time on the x axis. Velocity is defined as distance over time. When we calculate the area under the curve of our function over an interval. In this case our interval would be two points of time, and our area would be the distance travelled.
Is the area between two curves always positive?
Finally, unlike the area under a curve that we looked at in the previous chapter the area between two curves will always be positive. If we get a negative number or zero we can be sure that we’ve made a mistake somewhere and will need to go back and find it.
Why does the area under a curve become the anti-derivative?
However when it comes to the area under a curve for some reason when you break it up into an infinite amount of rectangles, magically it turns into the anti-derivative. Can someone explain why that is the definition of the integral and how Newton figured this out?
What is the area under the curve of velocity?
Velocity is defined as distance over time. When we calculate the area under the curve of our function over an interval. In this case our interval would be two points of time, and our area would be the distance travelled. In simple examples this is easy to do without thinking about the area under the curve or the integral.