JustAnswer



Find the Maclaurin series for f(x) using the definition of a Maclaurin series.Also find the associated radius of convergence.

f(x) = e ^-2x;

=[pic]

The above exponential series is valid for all values of x; hence, radius of convergence will be infinity.

f(x)=xcosx;

=[pic]

Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function.

f(x)=e^x + 2e^-x;

[pic]

f(x)=x² ln(1+3x³);

[pic]

Evaluate the indefinite integral as an infinite series.

∫[(e^x)-1/x]dx

[pic]

Find the Taylor polynomial T3 (x) for the function f centered at the number a.

f(x)=xcosx,a=0

(a)Approximate f by a Taylor polynomial with degree n a the number a.

=[pic]

(b)Use Taylor's Inequality to estimate the accuracy of the approximation f(x)≈Tn (x) when x lies in the given interval.

f(x)=x^-2, a=1, n=2, 0.9≤x≤1.1 Find (a) and (b).

=1-2(x-1)+3(x-1)2-………

T2(x) =f(x)-T2(x)

There exists ξ between 1 and x such that R2(x) = [pic].

= [pic]

|R2(x)| = [pic] in the interval [0.9, 1]

=[pic] in the interval [1,1.1]

Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which yhe curve is traced as t increases.

x=t², y=t³－4t, -3≦t≦3.

The following table gives values of t, x and y.

|t |x |y |

|-3 |9 |-15 |

|-2.5 |6.25 |-5.625 |

|-2 |4 |0 |

|-1.5 |2.25 |2.625 |

|-1 |1 |3 |

|-0.5 |0.25 |1.875 |

|0 |0 |0 |

|0.5 |0.25 |-1.875 |

|1 |1 |-3 |

|1.5 |2.25 |-2.625 |

|2 |4 |0 |

|2.5 |6.25 |5.625 |

|3 |9 |15 |

Plot is given below.

[pic]

(a)Eliminate the parameter to find a Cartesian equation of the curve.

y= t^3-4t=t(t^2-4) = t(x-4)

Or, y^2 = t^2(x-4)^2 = x(x-4)^2

Or, y^2 = x(x-4)^2 is the Cartesian equation.

Plot is given above.

(b)Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.

y=√t+1, y=√t-1 Find (a) and (b).

Kindly clarify x and y functions.

 Describe the motion of a particle with position(x,y) as t varies in the given interval.

x=2sint, y=4+cost, 0≦t≦3π/2.

| |t |x |y |

|0 |0 |0 |5 |

|30 |0.52359878 |1 |4.866025 |

|60 |1.04719755 |1.732051 |4.5 |

|90 |1.57079633 |2 |4 |

|120 |2.0943951 |1.732051 |3.5 |

|150 |2.61799388 |1 |3.133975 |

|180 |3.14159265 |2.45E-16 |3 |

|210 |3.66519143 |-1 |3.133975 |

|240 |4.1887902 |-1.73205 |3.5 |

|270 |4.71238898 |-2 |4 |

[pic]

sint = x/2 and cost = y-4

Hence, x^2/4 +(y-4)^2=sin2t+cos2t=1 is the Cartesian equation and plot is given above.

[pic]

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download