STUDY TOOL FOR THE STUDENTS WHO SEARCH NOTES BY ONLINE, Lecture notes of Earth science

Download STUDY TOOL FOR THE STUDENTS WHO SEARCH NOTES BY ONLINE and more Lecture notes Earth science in PDF only on Docsity! Understand Calculus Fundamentals of Calculus In this tutorial, we will cover the basics of calculus, including limits, derivatives, and integration. Limits Limits are used to evaluate a function when it is undefined or cannot be calculated at a certain value. By finding what happens to the function as x approaches that value, we can determine its behavior.  A limit expression can be used to evaluate the function at a certain value.  Factoring the function using the difference of squares method can help in evaluating the limit.  Limits can help us determine the behavior of a function as it approaches a certain value. Derivatives Derivatives give us the slope of the tangent line of a function at a certain value. The power rule is the most basic rule used to find the derivative of a function.  The power rule states that the derivative of a variable raised to a constant is n times x raised to the n-1 power.  The slope of the tangent line is equal to the derivative of the function at a certain value.  The slope of a secant line can be used to estimate the slope of the tangent line. Integration Integration is used to find the area under the curve of a function. It is the opposite of differentiation.  Integration is useful for calculating how much something accumulates over time.  The antiderivative or integral of a function is the reverse of its derivative. Slope of Tangent Line and Limits To approximate the slope of the tangent line, we can use the slope of the secant line with values closer to the point of interest. As the values get closer and closer, we can use limits to evaluate the slope of the tangent line. The formula for finding the slope of the tangent line using limits is: limx→a (f(x) - f(a)) / (x - a) To factor x3 - 8, we can use the formula for differences of perfect cubes, a3 - b3 = (a - b)(a2 + ab + b2). The derivative is a function that gives us the slope of the tangent line at some x value. The antiderivative or integration is the process of finding the function that accumulates over a period of time. Comparison of Derivatives and Antiderivatives Derivatives tell us the instantaneous rate of change, while antiderivatives or integration help us determine how much something accumulates over a period of time. Derivatives are used to calculate the slope of the tangent line, while integration is useful for calculating the area under the curve. In its simplest form, differentiation is dividing y values by x values, while integration is multiplying y values by x values. Example Problem Given the function a(t) = 0.01t2 + 0.5t + 100, we can calculate the amount of water in gallons in a tank at different times by plugging in values for t. To find how fast the amount of water is changing, we need to find the derivative of the function a(t) or a'(t). Using the power rule, we can find a'(t) = 0.02t + 0.5. To find how fast the amount of water is changing when t is 10, we plug in 10 into the derivative function and get a rate of change of 0.7 gallons per minute. To calculate the slope associated with derivatives, divide the y values by the x values. In this problem, the slope of the tangent line represents the instantaneous rate of change, while