Linear Approximation Examples Linear Approximation Examples Introduction By now we have seen many examples in which we determined the tangent line to the graph of a function f(x) at a point x = a. A linear approximation (or tangent line approximation) is the simple idea of using the equation of the tangent line to approximate values of f(x) for x near x = a. A picture really tells the whole story here. Take a look at the gure below in which the graph of a function f(x) is plotted along with its tangent line at x = a. Notice how, near the point of contact (a; f(a)), the tangent line nearly coincides with the graph of f(x), while the distance between the tangent line and graph grows as x moves away from a. Linear approximation is a part of calculus, which comes under mathematics. Here, we do the approximation of normal function with the help of Linear Function. It is mainly used in finite differences to solve the first order method to simplify the problem or approximate the result of the equation. The process to finding the straight line equation, y = mx + c, where m and c are constant is called as linear approximation. Know More About Mean Definition Math.Tutorvista.com
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Example 1 Determine the linear approximation for at . Use the linear approximation to approximate the value of and . Solution :- Since this is just the tangent line there really isn’t a whole lot to finding the linear approximation. The linear approximation is then, Now, the approximations are nothing more than plugging the given values of x into the linear approximation. For comparison purposes we’ll also compute the exact values. So, at this linear approximation does a very good job of approximating the actual value. However, at it doesn’t do such a good job. This shouldn’t be too surprising if you think about. Near both the function and the linear approximation have nearly the same slope and since they both pass through the point they should have nearly the same value as long as we stay close to . However, as we move away from the linear approximation is a line and so will always have the same slope while the function's slope will change as x changes and so the function will, in all likelihood, move away from the linear approximation. This approximation is crucial to many known numerical techniques such as Euler's Method to approximate solutions to ordinary differential equations. The idea to use linear approximations rests in the closeness of the tangent line to the graph of the function around a point. In other words, for a given value of x close to a, the dierence between the corresponding y value on the graph of f(x) and the y value on the tangent line is very small.
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The Product Rule The Product Rule Product rule is used to differentiate the function. When given function is the product of two or more functions,product rule is used.If the problems are combination of any two or more functions then their derivative can be find by using Product Rule. The derivative of a function h(x) will be denoted by D {h(x)} or h'(x). The product rule is a common rule for the differentiating problems where one function is multiplied by one another function.The derivative of the product of two differentiable functions is equal to the first function times the derivative of the second plus the second function times the derivative of the first function. The following problems require the use of the product rule. In the following discussion and solutions the derivative of a function h(x) will be denoted by or h'(x) . The product rule is a formal rule for differentiating problems where one function is multiplied by another. The rule follows from the limit definition of derivative and is given by Remember the rule in the following way. Each time, differentiate a different function in the product and add the two terms together. In the list of problems which follows, most problems are average and a few are somewhat challenging. In most cases, final answers to the following problems are given in the most simplified form.
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A good way to remember the product rule for differentiation is ``the first times the derivative of the second plus the second times the derivative of the first.'' It may seem non-intuitive now, but just see, and in a few days you'll be repeating it to yourself, too. Another way to remember the above derivation is to think of the product u(x)v(x) as the area of a rectangle with width u(x) and height v(x). The change in area is d(uv), and is indicated is the figure below. As x changes, the area changes from the area of the red rectangle, u(x)v(x), to the area of the largest rectangle, the sum of the read, green, blue and yellow rectangles. The change in area is the sum of the areas of the green, blue and yellow rectangles, In the limit of dx small, the area of the yellow rectangle is neglected. Algebraically, ``Neglecting'' the yellow rectangle is equivalent to invoking the continuity of u(x) above. This argument cannot constitute a rigourous proof, as it uses the differentials algebraically; rather, this is a geometric indication of why the product rule has the form it does. Example. Accepting for the moment that the derivative of sin x is cos x (Lesson 12), then Problem 3. Calculate the derivative of 5x sin x. 5x cos x + 5 sin x Proof of the product rule To prove the product rule, we will express the difference quotient simply as Δy Δx . (Lesson 5.) And so let y = f g. Then a change in y -- Δy -- will produce corresponding changes in f and g: Read More About Correlation Definition
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