Differentials Approximate Error
Contents |
available. Most of the classes have practice problems with solutions available on the practice problems pages. Also most classes have assignment using differentials to approximate error problems for instructors to assign for homework (answers/solutions to the assignment problems
Use Differentials To Approximate The Maximum Error
are not given or available on the site). Algebra [Notes] [Practice Problems] [Assignment Problems] Calculus I [Notes]
Differentials To Approximate Change
[Practice Problems] [Assignment Problems] Calculus II [Notes] [Practice Problems] [Assignment Problems] Calculus III [Notes] [Practice Problems] [Assignment Problems] Differential Equations [Notes] Extras Here are some extras topics that I
Differentials To Approximate Square Root
have on the site that do not really rise to the level of full class notes. Algebra/Trig Review Common Math Errors Complex Number Primer How To Study Math Close the Menu Current Location : Calculus I (Notes) / Applications of Derivatives / Differentials Calculus I [Notes] [Practice Problems] [Assignment Problems] Review [Notes] [Practice Problems] [Assignment Problems] Review : differentials to approximate volume Functions [Notes] [Practice Problems] [Assignment Problems] Review : Inverse Functions [Notes] [Practice Problems] [Assignment Problems] Review : Trig Functions [Notes] [Practice Problems] [Assignment Problems] Review : Solving Trig Equations [Notes] [Practice Problems] [Assignment Problems] Review : Trig Equations with Calculators, Part I [Notes] [Practice Problems] [Assignment Problems] Review : Trig Equations with Calculators, Part II [Notes] [Practice Problems] [Assignment Problems] Review : Exponential Functions [Notes] [Practice Problems] [Assignment Problems] Review : Logarithm Functions [Notes] [Practice Problems] [Assignment Problems] Review : Exponential and Logarithm Equations [Notes] [Practice Problems] [Assignment Problems] Review : Common Graphs [Notes] [Practice Problems] [Assignment Problems] Limits [Notes] [Practice Problems] [Assignment Problems] Tangent Lines and Rates of Change [Notes] [Practice Problems] [Assignment Problems] The Limit [Notes] [Practice Problems] [Assignment Problems] One-Sided Limits [Notes] [Practice Problems] [Assignment Problems] Limit Properties [Notes] [Practice Problems] [Assignment Problems] Computing Limits [Notes] [Practice Problems] [Assignment Problems] Infinite Limits [Notes] [Practice Problems] [Assignment Problems] Limits At Infinity, Part I [Notes] [Practice Problems] [Assignment Problems] Limits At Infinity, Part II [Notes] [Practice Problems] [Assignment Problems] Continuity [Notes] [Prac
Finite Math Everything for Finite Math & Calculus Español Note To understand this topic, you will need to be familiar with derivatives, as discussed in Chapter 3 of Calculus Applied to the Real World. If you like, you differentials to approximate cube root can review the topic summary material on techniques of differentiation or, for a more detailed study, use differentials to approximate the value of the expression the on-line tutorials on derivatives of powers, sums, and constant multipes. We start with the observation that if you zoom in to a use differentials to approximate the quantity portion of a smooth curve near a specified point, it becomes indistinguishable from the tangent line at that point. In other words: The values of the function are close to the values of the linear function whose graph is http://tutorial.math.lamar.edu/Classes/CalcI/Differentials.aspx the tangent line. For this reason, the linear function whose graph is the tangent line to $y = f(x)$ at a specified point $(a, f(a))$ is called the linear approximation of $f(x)$ near $x = a.$ Q What is the formula for the linear approximation? A All we need is the equation of the tangent line at a specified point $(a, f(a)).$ Since the tangent line at $(a, f(a))$ has slope $f'(a),$ we can write down its http://www.zweigmedia.com/RealWorld/calctopic1/linearapprox.html equation using the point-slope formula: $y= y_0 + m(x - x_0)$ $= f(a) + f'(a)(x - a)$ Thus, the the linear approximation to $f(x)$ near $x = a$ is given by $L(x) = f(a) + f'(a)(x - a).$ Q The above argument is based on geometry: the fact that the tangent line is close to the original graph near the point of tangency. Is there an algebriac way of seeing why this is true? A Yes. This links to an algebraic derivation of the linear approximation. Linear Approximation of $f(x)$ Near $x = a$ If $x$ is close to a, then $f(x) \approx f(a) + (x-a)f'(a).$ The right-hand side, $L(x) = f(a) + (x-a)f'(a),$ which is a linear function of $x,$ is called the linear approximation of $f(x)$ near $x = a.$ Example 1 Linear Approximation of the Square Root Let $f(x) = x^{1/2}.$ Find the linear approximation of $f$ near $x = 4$ (at the point $(4, f(4)) = (4, 2)$ on the graph), and use it to approximate $\sqrt{4.1.}$ Solution Since $f'(x) = 1/(2x^{1/2}),$ $f'(4) = 1/(2 \cdot 4^{1/2}) = 1/4.$ so the linear approximation is $L(x) = f(4) + (x-4)f'(4)$ $ = 2 + (x-4)/4$ $ = 0.25x + 1.$ We can use $L(x)$ to approximate the square root of any number close to $4$ very easily without using a
with: (1) Functions of several variables. (2) Evaluation of partial derivatives, and the chain rules of differentiation. (3) Manipulation of summations in algebraic https://www.lhup.edu/~dsimanek/scenario/errorman/calculus.htm context. At this mathematical level our presentation can be briefer. We can dispense with the tedious explanations and elaborations of previous chapters. 6.2 THE CHAIN RULE AND DETERMINATE ERRORS If a result R = R(x,y,z) is calculated from a number of data quantities, x, y and z, then the relation: [6-1] ∂R ∂R ∂R dR = differentials to —— dx + —— dy + —— dz ∂x ∂y ∂z
holds. This is one of the "chain rules" of calculus. This equation has as many terms as there are variables. Then, if the fractional errors are small, the differentials dR, dx, dy and dz may be replaced by the absolute errors ΔR, Δx, Δy, differentials to approximate and Δz, and written: [6-2] ∂R ∂R ∂R ΔR ≈ —— Δx + —— Δy + —— Δz ∂x ∂y ∂z Strictly this is no longer an equality, but an approximation to DR, since the higher order terms in the Taylor expansion have been neglected. So long as the errors are of the order of a few percent or less, this will not matter. This equation is now an error propagation equation. [6-3] Finally, divide equation (6.2) by R: ΔR x ∂R Δx y ∂R Δy z ∂R Δz —— = —————+——— ——+————— R R ∂x x R ∂y y R ∂z z The factors of the form Δx/x, Δy/y, etc are relative (fractional) errors. This equation shows how the errors in the result depend on the errors in the data. Eq. 6.2 and 6.3 are called the standard form error equations. They are also called determinate error equations, because they are strictly valid for determinate errors (not indeterminate errors). [We'll get to indeterminate error