The objective of this calculus lab is to explore and understand the concepts of linear and Taylor approximations for transcendental functions. Through hands-on exercises, students will:
By the end of this lab, students should be able to effectively use linear and Taylor approximations to estimate function values and understand the factors that influence the accuracy of these approximations.
A linear approximation estimates the value of a function near a given point using the tangent line at that point. It’s expressed as:
\[f(x) \approx f(a) + f'(a)(x - a)\]
where \(f(a)\) is the function value at point \(a\) and \(f'(a)\) is the derivative at point \(a\).
A Taylor approximation is a method for approximating a function \(f(x)\) near a point \(a\) using a polynomial. The \(n\)-th order Taylor polynomial of \(f\) around \(a\) is given by:
\[T_n(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x - a)^n\]
This polynomial approximates \(f(x)\) by matching its value and the values of its first \(n\) derivatives at \(a\).
Provide a linear approximation function for each of the transcendental functions listed below, centered around the point \(a\).
Evaluate the linear approximation at a point \(b\) close to the center of approximation.
Calculate the percentage error of the approximation using the formula: \[\boxed{ Error (\%) = \left( \frac{\text{approx} - \text{exact}}{\text{exact}} \right) \times 100 }\]
Linear Function \(L(x)\)
(%i1) a: 0;\[\mathtt{(\textit{%o}_{1})}\quad 0\]
(%i2) define(f(x),sin(x));\[\mathtt{(\textit{%o}_{2})}\quad f\left(x\right):=\sin x\]
(%i3) define(fp(x),diff(f(x),x,1));\[\mathtt{(\textit{%o}_{3})}\quad \textit{fp}\left(x\right):=\cos x\]
(%i4) define(L(x),f(a)+fp(a)*(x-a));\[\mathtt{(\textit{%o}_{4})}\quad L\left(x\right):=x\]
Error of approximation
(%i5) b: %pi/6;\[\mathtt{(\textit{%o}_{5})}\quad \frac{\pi}{6}\]
(%i6) L(b),numer;\[\mathtt{(\textit{%o}_{6})}\quad 0.5235987755982988\]
(%i7) sin(b),numer;\[\mathtt{(\textit{%o}_{7})}\quad 0.49999999999999994\]
(%i8) approx_error: (L(b)-sin(b))/(sin(b))*100,numer;\[\mathtt{(\textit{%o}_{8})}\quad 4.719755119659775\]
For the given list of functions, use the Taylor series to approximate each function up to the 7th order polynomial.
(%i9) a: 0;\[\mathtt{(\textit{%o}_{9})}\quad 0\]
(%i10) n: 7;\[\mathtt{(\textit{%o}_{10})}\quad 7\]
(%i11) define(f(x),sin(x));\[\mathtt{(\textit{%o}_{11})}\quad f\left(x\right):=\sin x\]
(%i12) taylor(f(x),x,a,n);\[\mathtt{(\textit{%o}_{12})}\quad -\left(\frac{x^7}{5040}\right)+\frac{x^5}{120}-\frac{x^3}{6}+x\]
For the given list of functions, follow these steps:
(%i13) a: 0;\[\mathtt{(\textit{%o}_{13})}\quad 0\]
(%i14) define(f(x),sin(x));\[\mathtt{(\textit{%o}_{14})}\quad f\left(x\right):=\sin x\]
(%i15) define(fp(x),diff(f(x),x,1))$
(%i16) define(L(x),f(a)+fp(a)*(x-a));\[\mathtt{(\textit{%o}_{16})}\quad L\left(x\right):=x\]
(%i17) plot2d([f(x),L(x),taylor(f(x),x,a,4),taylor(f(x),x,a,7)],[x,-%pi,%pi],[y,-5,5]);