Mathematics
Linear Approximations and Differentials
Linear approximations and differentials are mathematical tools used to estimate the value of a function near a specific point. The linear approximation is a straight-line approximation to a function, while the differential represents the change in the function's value due to a small change in the independent variable. These concepts are particularly useful in calculus for making quick and accurate estimations.
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8 Key excerpts on "Linear Approximations and Differentials"
eBook - PDFCalculus
Single Variable
- Carl V. Lutzer, H. T. Goodwill(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
Section 4.1 Linear Approximation 246 z Common Difficulties with Differentials Some students mistake dh for h 0 , but they are different things. Whereas h 0 tells us the instantaneous rate of change, the differential is a linear approximation of how much change happens. In the story of the model rocket, for example, the derivative is h 0 and has units of ft/sec, while the differential dh has units of feet. Units can help you reason your way to or through a formula. You should know • the terms linear approximation, linearization, Newton’s method, converge, diverge, and differential; • that Newton’s method is an algorithm for finding roots of functions; • why Newton’s method “fails” when h 0 (t) = 0; • that differentials give us a linear approximation of change; • that, graphically speaking, the differentials dy tells us about change along the tangent line, whereas Δy is the actual change in the function value. You should be able to • write the linear approximation of f (t) about t 0 ; • execute several steps of Newton’s method by hand; • use a linear approximation of f about t 0 to estimate f (t 0 + Δt); • write the Product Rule and Quotient Rule in terms of differentials; • use differentials to estimate error. z 4.1 Skill Exercises 1. Suppose f (2) = 3 and f 0 (2) = 7. Estimate f (2.001). 2. Suppose f (5) = 12 and f 0 (5) = 3. Estimate f (4.99). 3. Suppose f 0 (8) = 11. Approximate the change in f (t) when t is increased from 8 to 8.0001. 4. Suppose f 0 (-2) = 6. Approximate the change in f (t) when t is decreased from -2 to -2.17. Use linear approximation at t = 1 to estimate the numbers in #5–8. 5. f (1.01) when f (t) = 1 + t 9 6. f (0.9) when f (t) = 13t 8 7. f (0.9) when f (t) = ln(t) 8. f (1.1) when f (t) = sin(πt/6) In #9–12 use linearization at the specified t 0 to approximate the following numbers. 9. √ 24, t 0 = 25 10. 3 √ 65, t 0 = 64 11. cos(3), t 0 = π 12. e 0.1 , t 0 = 0
eBook - PDFCalculus
Early Transcendentals
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
3.5 Local Linear Approximation; Differentials 155 DIFFERENTIALS Up to now we have interpreted dy /dx as a single entity representing the derivative of y with respect to x; the symbols “dy” and “dx,” which are called differentials, have had no mean- ings attached to them. Our next goal is to define these symbols in such a way that dy /dx can be treated as an actual ratio. To do this, assume that f is differentiable at a point x, define dx to be an independent variable that can have any real value, and define dy by the formula dy = f (x) dx (5) If dx = 0, then we can divide both sides of (5) by dx to obtain dy dx = f (x) (6) This achieves our goal of defining dy and dx so their ratio is f (x). Formula (5) is said to express (6) in differential form. To interpret (5) geometrically, note that f (x) is the slope of the tangent line to the graph of f at x. The differentials dy and dx can be viewed as a corresponding rise and run of Figure 3.5.5 this tangent line (Figure 3.5.5). Example 3 Express the derivative with respect to x of y = x 2 in differential form, and discuss the relationship between dy and dx at x = 1. Solution. The derivative of y with respect to x is dy /dx = 2x, which can be expressed in differential form as dy = 2x dx When x = 1 this becomes dy = 2 dx This tells us that if we travel along the tangent line to the curve y = x 2 at x = 1, then a change of dx units in x produces a change of 2 dx units in y. Thus, for example, a run of Figure 3.5.6 dx = 2 units produces a rise of dy = 4 units along the tangent line (Figure 3.5.6). It is important to understand the distinction between the increment Δy and the differen- tial dy. To see the difference, let us assign the independent variables dx and Δx the same value, so dx = Δx.
eBook - PDFCalculus
Multivariable
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Local Linear Approximations We now show that if a function f is differentiable at a point, then it can be very closely approxi- mated by a linear function near that point. For example, suppose that f (x, y) is differentiable at the point (x 0 , y 0 ). Then approximation (3) can be written in the form f (x 0 + Δx, y 0 + Δy) ≈ f (x 0 , y 0 ) + f x (x 0 , y 0 )Δx + f y (x 0 , y 0 )Δy If we let x = x 0 + Δx and y = x 0 + Δy, this approximation becomes Show that if f (x, y) is a linear function, then (14) becomes an equality. f (x, y) ≈ f (x 0 , y 0 ) + f x (x 0 , y 0 )(x − x 0 ) + f y (x 0 , y 0 )(y − y 0 ) (14) which yields a linear approximation of f (x, y). Since the error in this approximation is equal to the error in approximation (3), we conclude that for (x, y) close to (x 0 , y 0 ), the error in (14) will be much smaller than the distance between these two points. When f (x, y) is differ- entiable at (x 0 , y 0 ) we let L(x, y) = f (x 0 , y 0 ) + f x (x 0 , y 0 )(x − x 0 ) + f y (x 0 , y 0 )(y − y 0 ) (15) and refer to L(x, y) as the local linear approximation to f at (x 0 , y 0 ). Explain why the error in approximation (14) is the same as the error in approx- imation (3). Example 4 Let L(x, y) denote the local linear approximation to f (x, y) = x 2 + y 2 at the point (3, 4). Compare the error in approximating f (3.04, 3.98) = (3.04) 2 + (3.98) 2 by L(3.04, 3.98) with the distance between the points (3, 4) and (3.04, 3.98). Solution We have f x (x, y) = x x 2 + y 2 and f y (x, y) = y x 2 + y 2 818 CHAPTER 13 Partial Derivatives with f x (3, 4) = 3 5 and f y (3, 4) = 4 5 . Therefore, the local linear approximation to f at (3, 4) is given by L(x, y) = 5 + 3 5 (x − 3) + 4 5 (y − 4) Consequently, f (3.04, 3.98) ≈ L(3.04, 3.98) = 5 + 3 5 (0.04) + 4 5 (−0.02) = 5.008 Since f (3.04, 3.98) = (3.04) 2 + (3.98) 2 ≈ 5.00819 the error in the approximation is about 5.00819 − 5.008 = 0.00019.
eBook - PDF- James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. SECTION 14.4 Tangent Planes and Linear Approximations 1017 ■ Differentials For a differentiable function of one variable, y - f s xd, we define the differential dx to be an independent variable; that is, dx can be given the value of any real number. The dif- ferential of y is then defined as 9 dy - f 9 s xd dx (See Section 2.9.) Figure 6 shows the relationship between the increment Dy and the differential dy : Dy represents the change in height of the curve y - f s xd and dy repre- sents the change in height of the tangent line when x changes by an amount dx - Dx. For a differentiable function of two variables, z - f s x, yd, we define the differentials dx and dy to be independent variables; that is, they can be given any values. Then the differential dz, also called the total differential, is defined by 10 dz - f x s x, yd dx 1 f y s x, yd dy - -z -x dx 1 -z -y dy (Compare with Equation 9.) Sometimes the notation df is used in place of dz. If we take dx - Dx - x 2 a and dy - Dy - y 2 b in Equation 10, then the differen- tial of z is dz - f x sa, bds x 2 ad 1 f y sa, bds y 2 bd So, in the notation of differentials, the linear approximation (4) can be written as f s x, yd < f sa, bd 1 dz Figure 7 is the three-dimensional counterpart of Figure 6 and shows the geometric inter- pretation of the differential dz and the increment Dz: dz represents the change in height of the tangent plane, whereas Dz represents the change in height of the surface z - f s x, yd when s x, yd changes from sa, bd to sa 1 Dx, b 1 Dyd. y x z Îx=dx 0 {a, b, f(a, b)} (a, b, 0) (a+Îx, b+Îy, 0) {a+Îx, b+Îy, f(a+Îx, b+Îy)} f(a, b) f(a, b) Îy=dy tangent plane z-f(a, b)=f x (a, b)(x-a)+f y (a, b)( y-b) surface z=f(x, y) dz Îz EXAMPLE 4 (a) If z - f s x, yd - x 2 1 3xy 2 y 2 , find the differential dz.
eBook - PDFAnton's Calculus
Early Transcendentals
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
Explain your answer. 47. A differential dy is defined to be a very small change in y. 48. The error in approximation (2) is the same as the error in approximation (7). 49. A local linear approximation to a function can never be identically equal to the function. 50. A local linear approximation to a nonconstant function can never be constant. 51–54 Use the differential dy to approximate Δy when x changes as indicated. 51. y = √ 5x − 1; from x = 2 to x = 2.03 52. y = √ x 3 + 8; from x = 1 to x = 0.97 53. y = 1 x 2 + 1 ; from x = 2 to x = 1.96 54. y = x √ 7x + 4; from x = 3 to x = 3.05 55. The side of a square is measured to be 10 ft, with a possible error of ±0.1 ft. (a) Use differentials to estimate the error in the calculated area. (b) Estimate the percentage errors in the side and the area. 56. The side of a cube is measured to be 25 cm, with a possible error of ±1 cm. (a) Use differentials to estimate the error in the calculated volume. (b) Estimate the percentage errors in the side and volume. 57. The hypotenuse of a right triangle is known to be 10 in exactly, and one of the acute angles is measured to be 30 ◦ , with a possible error of ±1 ◦ . (a) Use differentials to estimate the errors in the sides opposite and adjacent to the measured angle. (b) Estimate the percentage errors in the sides. 58. One side of a right triangle is known to be 25 cm exactly. The angle opposite to this side is measured to be 60 ◦ , with a possible error of ±0.5 ◦ . (a) Use differentials to estimate the errors in the adjacent side and the hypotenuse. (cont.) 160 Chapter 3 / Topics in Differentiation (b) Estimate the percentage errors in the adjacent side and hypotenuse. 59. The electrical resistance R of a certain wire is given by R = k ∕ r 2 , where k is a constant and r is the radius of the wire. Assuming that the radius r has a possible error of ±5%, use differentials to estimate the percentage error in R.
- James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. SECTION 14.4 Tangent Planes and Linear Approximations 979 ■ Differentials For a differentiable function of one variable, y - f s xd, we define the differential dx to be an independent variable; that is, dx can be given the value of any real number. The dif- ferential of y is then defined as 9 dy - f 9 s xd dx (See Section 3.10.) Figure 6 shows the relationship between the increment Dy and the differential dy : Dy represents the change in height of the curve y - f s xd and dy repre- sents the change in height of the tangent line when x changes by an amount dx - Dx. For a differentiable function of two variables, z - f s x, yd, we define the differentials dx and dy to be independent variables; that is, they can be given any values. Then the differential dz, also called the total differential, is defined by 10 dz - f x s x, yd dx 1 f y s x, yd dy - -z -x dx 1 -z -y dy (Compare with Equation 9.) Sometimes the notation df is used in place of dz. If we take dx - Dx - x 2 a and dy - Dy - y 2 b in Equation 10, then the differen- tial of z is dz - f x sa, bds x 2 ad 1 f y sa, bds y 2 bd So, in the notation of differentials, the linear approximation (4) can be written as f s x, yd < f sa, bd 1 dz Figure 7 is the three-dimensional counterpart of Figure 6 and shows the geometric inter- pretation of the differential dz and the increment Dz: dz represents the change in height of the tangent plane, whereas Dz represents the change in height of the surface z - f s x, yd when s x, yd changes from sa, bd to sa 1 Dx, b 1 Dyd. y x z Îx=dx 0 {a, b, f(a, b)} (a, b, 0) (a+Îx, b+Îy, 0) {a+Îx, b+Îy, f(a+Îx, b+Îy)} f(a, b) f(a, b) Îy=dy tangent plane z-f(a, b)=f x (a, b)(x-a)+f y (a, b)( y-b) surface z=f(x, y) dz Îz EXAMPLE 4 (a) If z - f s x, yd - x 2 1 3xy 2 y 2 , find the differential dz.
eBook - ePub- Cinzia Bisi, Rita Fioresi(Authors)
- 2024(Publication Date)
- Chapman and Hall/CRC(Publisher)
3 Applications of the DerivativeDOI: 10.1201/9781003343288-33.1 The Linear Approximation
The concept of linear approximation is based on the geometric interpretation of the derivative. In fact, as we saw in Chapter 2 , we can view the derivative of a function f at a point, as the slope of the tangent line to the curve y = f(x) at that point. Consider a function f : D → ℝ and two points of the domain P and Q, with coordinates (x0 , f(x0 )) and (x0 + h, f(x0 + h)), respectively. We see that the line passing through the points P and Q approximates the tangent line to graph y = f(x) and also the graph itself. In particular, we get a good approximation, by choosing h very small. We emphasize once again that this is just an intuitive reasoning: only the definition of limit can rigorously account for such notions such as “near”, “approximate” and so on.As we see from the graph, the difference quotientf (hx 0+ h ) − f (x 0)represents the slope of the line passing through P and Q. If we consider a “small” value of h, we can effectively approximate the derivative asThe symbol “≃” means that we have an approximation and not an equality.f ′(x 0) ≃f (hx 0+ h ) − f (x 0)We obtain an estimate for the value of f(x0 + h), which is called linear approximation of f:We can also express it more concisely as:f ((3.1)x 0+ h ) ≃ f (x 0) +f ′(x 0) ⋅ h .Δ ( f ) ≃(3.2)f ′(x 0) Δ x , where Δ ( f ) = f (x 0+ h ) − f (x 0) , Δ x = h .For example, if we consider the function f(x) = 2x2 , for x0 = 1 and h = 0.02, we can write the formula (3.1) as:f ( 1.02 ) ≅ f ( 1 ) +f ′( 1 ) ⋅ 0.02 = 2 + 4 ⋅ 1 ⋅ 0.02 = 2.08Checking with a calculator: f(1.02) = 2 · 1.022 = 2.0808. If we choose a larger value of h, for example, h
eBook - PDFCalculus
Multivariable
- Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
842 Chapter 14 DIFFERENTIATING FUNCTIONS OF SEVERAL VARIABLES Notice that the linear terms in (, ) are the same as the linear terms in (, ). The quadratic terms in (, ) can be thought of as “correction terms” to the linear approximation. (b) The contour plots of (, ), (, ), and (, ) are in Figures 14.58–14.60. 1 −1 1 −1 2 1 0 −1 −2 Figure 14.58: Original function, (, ) 1 −1 1 −1 2 1 0 −1 −2 Figure 14.59: Linear approximation, (, ) 1 −1 1 −1 2 1 0 −1 −2 Figure 14.60: Quadratic approximation, (, ) Notice that the contour plot of is more similar to the contour plot of than is the contour plot of . Since is linear, the contour plot of consists of parallel, equally spaced lines. An alternative, and much quicker, way to find the Taylor polynomial in the previous example is to use the single-variable approximations. For example, since cos = 1 − 2 2! + 4 4! + ⋯ and sin = − 3 3! + ⋯ , we can substitute = 2 + and = + and expand. We discard terms beyond the second (since we want the quadratic polynomial), getting cos(2 + ) = 1 − (2 + ) 2 2! + (2 + ) 4 4! + ⋯ ≈ 1 − 1 2 (4 2 + 4 + 2 ) = 1 − 2 2 − 2 − 1 2 2 and sin( + ) = ( + ) − ( + ) 3 3! + ⋯ ≈ + . Combining these results, we get cos(2 + ) + 3 sin( + ) ≈ 1 − 2 2 − 2 − 1 2 2 + 3( + ) = 1 + 3 + 3 − 2 2 − 2 − 1 2 2 . Linear and Quadratic Approximations Near (, ) The local linearization for a function (, ) at a point (, ) is Taylor Polynomial of Degree Approximating (, ) for (, ) Near (, ) If has continuous first-order partial derivatives, then (, ) ≈ (, ) = (, ) + (, )( − ) + (, )( − ). 14.7 SECOND-ORDER PARTIAL DERIVATIVES 843 This suggests that a quadratic polynomial approximation (, ) for (, ) near a point (, ) should be written in terms of ( − ) and ( − ) instead of and .
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