Express the vector 1 2 3 5 with respect to this basis

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Unformatted text preview: ∞ lim (1 + 1x ) = e(limx→∞ x 1 x ln(1+ x )) > Hint(%); [lhopital , ln(1 + > Rule[%](%%); 1 )] x x→∞ lim (1 + 1x ) = e(limx→∞ x x ) x+1 92 • Chapter 4: Maple Organization > Hint(%); rewrite , > Rule[%](%%); x = x+1 1 1 1+ x x→∞ lim (1 + 1x ) =e x Plotting a Function and A Tangent Line Consider the function −2/3 ∗ x2 + x. Plot the function and its tangent line at x = 0. > Tangent(-2/3*x^2+x, x=0, -2..2, output=plot, > showtangent=true); TheTangenttotheGraphof f(x)=–2/3*x^2+x atthePoint(0,f(0)) 2 1 –2 –1 x 1 2 –1 –2 –3 –4 –5 f(x) Thetangentatx=0 Where does this curve cross the x-axis? > Roots(-2/3*x^2+x); [0, 3 ] 2 4.2 The Maple Packages • 93 You can find the area under the curve between these two points by using Riemann sums. > ApproximateInt(-2/3*x^2+x, x=0..3/2, method=midpoint, > output=plot, view=[0..1.6, -0.15..0.4]); AnApproximationoftheIntegralof f(x)=–2/3*x^2+x ontheInterval[0,3/2] UsingaMidpointRiemannSum ApproximateValue:.3750000000 0.4 0.3 0.2 0.1 0 –0.1 0.2 0.4 0.6 0.8 x 1 1.2 1.4 1.6 –0.2 Area:.3768750000 f(x) Since the result is not a good approximation, increase the number of boxes used to forty. > ApproximateInt(-2/3*x^2+x, x=0..3/2, method=midpoint, > output=plot, view=[0..1.6, -0.15..0.4], > partition=40); AnApproximationoftheIntegralof f(x)=–2/3*x^2+x ontheInterval[0,3/2] UsingaMidpointRiemannSum ApproximateValue:.3750000000 0.4 0.3 0.2 0.1 0 –0.1 0.2 0.4 0.6 0.8 x 1 1.2 1.4 1.6 –0.2 Area:.3751171867 f(x) To determine the actual value, take the limit as n goes to ∞. Use n boxes and change the output to sum . > ApproximateInt(-2/3*x^2+x, x=0..3/2, method=midpoint, > output=sum, partition=n); 94 • Chapter 4: Maple Organization n−1 i=0 3 2 12 1 (i + ) i+ 3 2 +3 2 − 2 n2 2n n Take the limit as n goes to ∞. > Limit( %, n=infinity ); n−1 i=0 3 n→∞ 2 lim > value(%); 12 1 (i + ) i+ 3 2 +3 2 − 2 n2 2n n 3 8 Observe that you can obtain the same result by using an integral. > Int(-2/3*x^2+x, x=0..3/2 ); 3/2 − 0 22 x + x dx 3 > value(%); 3 8 For more information on calculus with Maple, see chapter 7. The LinearAlgebra Package The LinearAlgebra package contains routines for computational linear algebra. • For a complete list of commands, refer to the ?LinearAlgebra help page. • For the student version, refer to the ?Student[LinearAlgebra] help page. 4.2 The Maple Packages • 95 The following examples are generated using the LinearAlgebra package. In linear algebra, a set of linearly independent vectors that generates a vector space is a basis. That is, you can uniquely express any element in the vector space as a linear combination of the elements of the basis. A set of vectors {v1 , v2 , v3 , . . . , vn } is linearly independent if and only if c1 v1 + c2 v2 + c3 v3 + · · · + cn vn = 0 implies c1 = c2 = c3 = · · · = cn = 0. Determining a Basis Determine a basis for the vector space generated by the vecto...
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