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APEX Calculus

Section 9.7 Taylor Polynomials

Figure 9.7.1. Video introduction to Section 9.7
Consider a function y=f(x) and a point (c,f(c)). The derivative, f(c), gives the instantaneous rate of change of f at x=c. Of all lines that pass through the point (c,f(c)), the line that best approximates f at this point is the tangent line; that is, the line whose slope (rate of change) is f(c).
In Figure 9.7.2, we see a function y=f(x) graphed. The table in Figure 9.7.3 shows that f(0)=2 and f(0)=1; therefore, the tangent line to f at x=0 is p1(x)=1(x0)+2=x+2. The tangent line is also given in the figure. Note that “near” x=0, p1(x)f(x); that is, the tangent line approximates f well.
The graph of a generic function is shown, along with the tangent line to the graph at x=0.
The image shows the graph of a function f(x) and its tangent line at (0,f(0)). The precise features of the graph of f(x) are unimportant for this illustration. What is relevant is that points on the tangent line, which is the graph of a linear function labeled as p1(x), are close to the corresponding points on the graph of f(x), near the point (0,f(0)). In other words, the image illustrates the fact that when x is close to zero, the value of p1(x) is close to the value of f(x).
Figure 9.7.2. A graph of f(x) and its tangent line at 0
f(0)=2 f(0)=1
f(0)=1 f(4)(0)=12
f(0)=2 f(5)(0)=19
Figure 9.7.3. Derivatives of f evaluated at 0
One shortcoming of this approximation is that the tangent line only matches the slope of f; it does not, for instance, match the concavity of f. We can find a polynomial, p2(x), that does match the concavity near 0 without much difficulty, though. The table in Figure 9.7.3 gives the following information:
f(0)=2f(0)=1f(0)=2.
Therefore, we want our polynomial p2(x) to have these same properties. That is, we need
p2(0)=2p2(0)=1p2(0)=2.
This is simply an initial-value problem. We can solve this using the techniques first described in Section 5.1. To keep p2(x) as simple as possible, we’ll assume that not only p2(0)=2, but that p2(x)=2. That is, the second derivative of p2 is constant, meaning p2 is a quadratic function.
If p2(x)=2, then p2(x)=2x+C for some constant C. Since we have determined that p2(0)=1, we find that C=1 and so p2(x)=2x+1. Finally, we can compute p2(x)=x2+x+C. Using our initial values, we know p2(0)=2 so C=2. We conclude that p2(x)=x2+x+2. This function is plotted with f in Figure 9.7.4.
We can repeat this approximation process by creating polynomials of higher degree that match more of the derivatives of f at x=0. In general, a polynomial of degree n can be created to match the first n derivatives of f. Figure 9.7.4 shows p4(x)=x4/2x3/6+x2+x+2, whose first four derivatives at 0 match those of f. (Using the table in Figure 9.7.3, start with p4(4)(x)=12 and solve the related initial-value problem.)
The graph of a function and two polynomials that approximate the function near x=0.
The graph of a function f(x) is shown. It is the same function as the first image in this section, but again, the precise details of the graph are unimportant.
Also shown are the graphs of two functions p2(x) and p4(x). The function p2(x) is quadratic, and its graph is a parabola that opens upward. The function p4(x) is a polynomial of degree 4.
All three graphs intersect at the point (0,f(0)), and the values of both p2(x) and p4(x) are close to the value of f(x) when x is close to 0. Two observations are important: first, both of these polynomial graphs appear to lie more closely to the graph of f(x) than the tangent line in the first image. Second, the graph of p4(x) is a good approximation to f(x) over a larger interval than the graph of p2(x).
In particular, the point (0,f(0)) appears to be a local minimum, and there is a corresponding minimum in the graphs of both p2(x) and p4(x). But the graph of f(x) also appears to have a local maximum near x=1. Near x=1, the graph of p2(x) separates from that of f(x): the first continues to increase, while the second begins to decrease. Near x=1, p2(x) is no longer a good approximation to f(x).
However, the graph of p4(x) also has a maximum near x=1, and p4(x) appears to be a good approximation to f(x) at least until x=2.
Figure 9.7.4. Plotting f, p2 and p4
Figure 9.7.5. Determining the coefficients of a Taylor polynomial
As we use more and more derivatives, our polynomial approximation to f gets better and better. In this example, the interval on which the approximation is “good” gets bigger and bigger. Figure 9.7.6 shows p13(x); we can visually affirm that this polynomial approximates f very well on [2,3]. The polynomial p13(x) is not particularly “nice”. It is
p13(x)=16901x136227020800+13x1212096001321x1139916800779x101814400359x9362880+x8240+139x75040+11x636019x5120x42x36+x2+x+2.
The graph of a function is shown, along with the graph of a degree 13 polynomial that closely approximates the function.
The graph of a function f(x) is shown. It is the same function as in the previous two images. Also shown is the graph of a degree 13 polynomial p13(x). We can see that the values of f(x) and p13(x) are very close over an interval from 2 to 3. For x<3, the graph of f(x) appears to approach a horizontal asymptote, while the graph of p13(x) approaches , so it ceases to be a good approximation to f(x) at this point.
Figure 9.7.6. Plotting f and p13
The polynomials we have created are examples of Taylor polynomials, named after the British mathematician Brook Taylor who made important discoveries about such functions. While we created the above Taylor polynomials by solving initial-value problems, it can be shown that Taylor polynomials follow a general pattern that make their formation much more direct. This is described in the following definition.

Definition 9.7.7. Taylor Polynomial, Maclaurin Polynomial.

Let f be a function whose first n derivatives exist at x=c.
  1. The Taylor polynomial of degree n of f at x=c is
    pn(x)=f(c)+f(c)(xc)+f(c)2!(xc)2+f(c)3!(xc)3++f(n)(c)n!(xc)n.
  2. A special case of the Taylor polynomial is the Maclaurin polynomial, where c=0. That is, the Maclaurin polynomial of degree n of f is
    pn(x)=f(0)+f(0)x+f(0)2!x2+f(0)3!x3++f(n)(0)n!xn.
Figure 9.7.8. Video presentation of Definition 9.7.7
We will practice creating Taylor and Maclaurin polynomials in the following examples.

Example 9.7.9. Finding and using Maclaurin polynomials.

  1. Find the nth Maclaurin polynomial for f(x)=ex.
  2. Use p5(x) to approximate the value of e.
Solution 1.
  1. We start with creating a table of the derivatives of ex evaluated at x=0. In this particular case, this is relatively simple, as shown in Figure 9.7.10.
    f(x)=ex f(0)=1
    f(x)=ex f(0)=1
    f(x)=ex f(0)=1
    f(n)(x)=ex f(n)(0)=1
    Figure 9.7.10. The derivatives of f(x)=ex evaluated at x=0
    By the definition of the Maclaurin polynomial, we have
    pn(x)=f(0)+f(0)x+f(0)2!x2+f(0)3!x3++f(n)(0)n!xn=1+x+12x2+16x3+124x4++1n!xn.
  2. Using our answer from part 1, we have
    exp5(x)=1+x+12x2+16x3+124x4+1120x5.
    To approximate the value of e, note that e=e1=f(1)p5(1). It is very straightforward to evaluate p5(1):
    p5(1)=1+1+12+16+124+1120=163602.71667.
    A plot of f(x)=ex and p5(x) is given in Figure 9.7.11. To 5 decimal places, the actual value of e is 2.71828. So this approximation agrees to two decimal places.
The graph of the exponential function and its degree 5 Maclaurin polynomial.
The graph of f(x)=ex is shown on the interval [3,2]. Also shown is the graph of p5(x), the degree 5 Maclaurin polynomial of f(x). For 2<x<2, there is very little visible difference between the two graphs. Near x=3 we can see that the two graphs begin to separate.
Figure 9.7.11. A plot of f(x)=ex and its 5th degree Maclaurin polynomial p5(x)
Solution 2. Video solution

Example 9.7.12. Finding and using Taylor polynomials.

  1. Find the nth Taylor polynomial of y=ln(x) at x=1.
  2. Use p6(x) to approximate the value of ln(1.5).
  3. Use p6(x) to approximate the value of ln(2).
Solution 1.
  1. We begin by creating a table of derivatives of ln(x) evaluated at x=1. While this is not as straightforward as it was in the previous example, a pattern does emerge (for n1), as shown in Figure 9.7.13. Notice in the table below that each time we take a derivative (starting at the second derivative), we apply the power rule and “bring down” the exponent to multiply by the previous coefficent. So the 6 in the 4th derivative is actually 123=3!.
    f(x)=ln(x) f(1)=0
    f(x)=1x f(1)=1
    f(x)=1x2 f(1)=1
    f(x)=2x3 f(1)=2
    f(4)(x)=6x4 f(4)(1)=6
    f(n)(x)= f(n)(1)=
    (1)n+1(n1)!xn (1)n+1(n1)!
    Figure 9.7.13. Derivatives of ln(x) evaluated at x=1
    Notice that the coefficients alternate in sign starting at n=1. Using Definition 9.7.7, we have
    pn(x)=f(c)+f(c)(xc)+f(c)2!(xc)2+f(c)3!(xc)3++f(n)(c)n!(xc)n=0+0!1!(x1)1!2!(x1)2+2!3!(x1)3++(1)n+1(n1)!n!(x1)n=(x1)12(x1)2+13(x1)314(x1)4++(1)n+1n(x1)n.
    Note how the coefficients of the (x1) terms turn out to be “nice.”
  2. We can compute p6(x) using our work above:
    p6(x)=(x1)12(x1)2+13(x1)314(x1)4+15(x1)516(x1)6.
    Since p6(x) approximates ln(x) well near x=1, we approximate ln(1.5)p6(1.5):
    p6(1.5)=(1.51)12(1.51)2+13(1.51)3+14(1.51)4+15(1.51)516(1.51)6=2596400.404688.
    This is a good approximation as a calculator shows that ln(1.5)0.4055. Figure 9.7.14 below plots y=ln(x) with y=p6(x). We can see that ln(1.5)p6(1.5).
  3. We approximate ln2 with p6(2):
    p6(2)=(21)12(21)2+13(21)314(21)4++15(21)516(21)6=112+1314+1516=37600.616667.
    This approximation is not terribly impressive: a hand held calculator shows that ln(2)0.693147. The graph in Figure 9.7.14 shows that p6(x) provides less accurate approximations of ln(x) as x gets close to 0 or 2. Surprisingly enough, even the 20th degree Taylor polynomial fails to approximate ln(x) for x>2 very well, as shown in Figure 9.7.15. We’ll soon discuss why this is.
A graph of the natural logarithm function and its degree 6 Taylor polynomial centered at x=1.
The graph of y=ln(x) is shown on the interval (0,3). Also shown is the graph of p6(x), the degree 6 Taylor polynomial of ln(x), centered at x=1. The graph of p6(x) is very close to the graph of ln(x) on the interval (0.5,1.5), but does not approximate the logarithm very well outside of this interval.
Figure 9.7.14. A plot of y=ln(x) and its 6th degree Taylor polynomial at x=1
A graph of the natural logarithm function and its degree 20 Taylor polynomial centered at x=1.
The graph y=ln(x) is shown again, this time with the degree 20 Taylor polynomial of ln(x), centered at x=1. Unlike with the example involving y=ex, increasing the degree of the Taylor polynomial did not do much to improve how well it approximates the logarithm. The approximation appears to be accurate over a slightly larger interval than the degree 6 approximation, but there is not a significant difference between the two images.
Figure 9.7.15. A plot of y=ln(x) and its 20th degree Taylor polynomial at x=1
Solution 2. Video solution
Taylor polynomials are used to approximate functions f(x) in mainly two situations:
  1. When f(x) is known, but perhaps “hard” to compute directly. For instance, we can define the cosine of an angle as either the ratio of sides of a right triangle (“adjacent over hypotenuse”) or using the definition in terms of the unit circle. However, neither of these provides a convenient way of computing cos(2). A Taylor polynomial of sufficiently high degree can provide a reasonable method of computing such values using only operations usually hard-wired into a computer (+, , × and ÷).
  2. When f(x) is not known, but information about its derivatives is known. This occurs more often than one might think, especially in the study of differential equations.
In both situations, a critical piece of information to have is “How good is my approximation?” If we use a Taylor polynomial to compute cos(2), how do we know how accurate the approximation is?
We had the same problem when studying Numerical Integration. Theorem 5.5.24 provided bounds on the error when using, say, Simpson’s Rule to approximate a definite integral. These bounds allowed us to determine that, for instance, using 10 subintervals provided an approximation within ±0.01 of the exact value. The following theorem gives similar bounds for Taylor (and hence Maclaurin) polynomials.
Figure 9.7.17. Video presentation of Theorem 9.7.16
The first part of Taylor’s Theorem states that f(x)=pn(x)+Rn(x), where pn(x) is the nth order Taylor polynomial and Rn(x) is the remainder, or error, in the Taylor approximation. The second part gives bounds on how big that error can be. If the (n+1)th derivative is large on I, the error may be large; if x is far from c, the error may also be large. However, the (n+1)! term in the denominator tends to ensure that the error gets smaller as n increases.
The following example computes error estimates for the approximations of ln(1.5) and ln(2) made in Example 9.7.12.

Example 9.7.18. Finding error bounds of a Taylor polynomial.

Use Theorem 9.7.16 to find error bounds when approximating ln(1.5) and ln(2) with p6(x), the Taylor polynomial of degree 6 of f(x)=ln(x) at x=1, as calculated in Example 9.7.12.
Solution 1.
  1. We start with the approximation of ln(1.5) with p6(1.5). The theorem references an open interval I that contains both x and c. The smaller the interval we use the better; it will give us a more accurate (and smaller!) approximation of the error. We let I=(0.9,1.6), as this interval contains both c=1 and x=1.5. The theorem references max|f(n+1)(z)|. In our situation, this is asking “How big can the 7th derivative of y=ln(x) be on the interval (0.9,1.6)?” The seventh derivative is y=6!/x7. The largest absolute value it attains on I is about 1506. (There are no critical numbers of f(7) in the interval so we evaluate the endpoints: f(7)(0.9)1506 and f(7)(1.6)27.) In particular, we are evaluating at x=1.5, so we let x=1.5. Thus we can bound the error as:
    |R6(1.5)|max|f(7)(z)|7!|(1.51)7|150650401270.0023.
    We computed p6(1.5)=0.404688; using a calculator, we find ln(1.5)0.405465, so the actual error is about 0.000778, which is less than our bound of 0.0023. This affirms Taylor’s Theorem; the theorem states that our approximation would be within about 2 thousandths of the actual value, whereas the approximation was actually closer. Taylor’s Theorem only gives an upper bound on the error.
  2. We again find an interval I that contains both c=1 and x=2; we choose I=(0.9,2.1). The maximum value of the seventh derivative of f on this interval is again about 1506 (as the largest values come near x=0.9). Thus
    |R6(2)|max|f(7)(z)|7!|(21)7|15065040170.30.
    This bound is not as nearly as good as before. Using the degree 6 Taylor polynomial at x=1 will bring us within 0.3 of the correct answer. As p6(2)0.61667, our error estimate guarantees that the actual value of ln(2) is somewhere between 0.31667 and 0.91667. These bounds are not particularly useful. In reality, our approximation was only off by about 0.07. However, we are approximating ostensibly because we do not know the real answer. In order to be assured that we have a good approximation, we would have to resort to using a polynomial of higher degree.
Solution 2. Video solution
We practice again. This time, we use Taylor’s theorem to find n that guarantees our approximation is within a certain amount.

Example 9.7.19. Finding sufficiently accurate Taylor polynomials.

Find n such that the nth Taylor polynomial of f(x)=cos(x) at x=0 approximates cos(2) to within 0.001 of the actual answer. What is pn(2)?
Solution 1.
Following Taylor’s theorem, we need bounds on the size of the derivatives of f(x)=cos(x). In the case of this trigonometric function, this is easy. All derivatives of cosine are ±sin(x) or ±cos(x). In all cases, these functions are never greater than 1 in absolute value. We want the error to be less than 0.001. To find the appropriate n, consider the following inequalities:
max|f(n+1)(z)|(n+1)!|(20)(n+1)|0.0011(n+1)!2(n+1)0.001.
We find an n that satisfies this last inequality with trial-and-error. When n=8, we have 28+1(8+1)!0.0014; when n=9, we have 29+1(9+1)!0.000282<0.001. Thus we want to approximate cos(2) with p9(2).
We now set out to compute p9(x). We again need a table of the derivatives of f(x)=cos(x) evaluated at x=0. A table of these values is given in Figure 9.7.20.
f(x)=cos(x) f(0)=1
f(x)=sin(x) f(0)=0
f(x)=cos(x) f(0)=1
f(x)=sin(x) f(0)=0
f(4)(x)=cos(x) f(4)(0)=1
f(5)(x)=sin(x) f(5)(0)=0
f(6)(x)=cos(x) f(6)(0)=1
f(7)(x)=sin(x) f(7)(0)=0
f(8)(x)=cos(x) f(8)(0)=1
f(9)(x)=sin(x) f(9)(0)=0
Figure 9.7.20. A table of the derivatives of f(x)=cos(x) evaluated at x=0
Notice how the derivatives, evaluated at x=0, follow a certain pattern. All the odd powers of x in the Taylor polynomial will disappear as their coefficient is 0. While our error bounds state that we need p9(x), our work shows that this will be the same as p8(x).
Since we are forming our polynomial at x=0, we are creating a Maclaurin polynomial, and:
p8(x)=f(0)+f(0)x+f(0)2!x2+f(0)3!x3++f(8)(0)8!x8=112!x2+14!x416!x6+18!x8.
We finally approximate cos(2):
cos(2)p8(2)=1313150.41587.
Our error bound guarantee that this approximation is within 0.001 of the correct answer. Technology shows us that our approximation is actually within about 0.0003 of the correct answer.
Figure 9.7.21 shows a graph of y=p8(x) and y=cos(x). Note how well the two functions agree on about (π,π).
A graph of the cosine function is shown, along with its degree 8 Maclaurin polynomial approximation.
The graph y=cos(x) is given on the interval [5,5], along with the graph of p8(x), the degree 8 Maclaurin polynomial approximation of cos(x).
The image illustrates how well the Maclaurin polynomials approximate the cosine function. With a degree 8 polynomial, there is little to no visible difference between the graphs over the interval (π,π).
Figure 9.7.21. A graph of f(x)=cos(x) and its degree 8 Maclaurin polynomial
Solution 2. Video solution

Example 9.7.22. Finding and using Taylor polynomials.

  1. Find the degree 4 Taylor polynomial, p4(x), for f(x)=x at x=4.
  2. Use p4(x) to approximate 3.
  3. Find bounds on the error when approximating 3 with p4(3).
Solution.
  1. We begin by evaluating the derivatives of f at x=4. This is done in Figure 9.7.23.
    f(x)=x f(4)=2
    f(x)=12x f(4)=14
    f(x)=14x3/2 f(4)=132
    f(x)=38x5/2 f(4)=3256
    f(4)(x)=1516x7/2 f(4)(4)=152048
    Figure 9.7.23. A table of the derivatives of f(x)=x evaluated at x=4
    These values allow us to form the Taylor polynomial p4(x):
    p4(x)=2+14(x4)+1/322!(x4)2+3/2563!(x4)3+15/20484!(x4)4.
  2. As p4(x)x near x=4, we approximate 3 with p4(3)=1.73212.
  3. To find a bound on the error, we need an open interval that contains x=3 and x=4. We set I=(2.9,4.1). The largest value the fifth derivative of f(x)=x takes on this interval is near x=2.9, at about 0.0273. (We often graph the (n+1)th derivative to find its extrema. In this case is f(5)(x)=105/(32x9/2) is always decreasing, so the maximum occurs at 2.9.) Thus
    |R4(3)|0.02735!|(34)5|0.00023.
    This shows our approximation is accurate to at least the first 2 places after the decimal. (It turns out that our approximation is actually accurate to 4 places after the decimal.) A graph of f(x)=x and p4(x) is given in Figure 9.7.24. Note how the two functions are nearly indistinguishable on (2,7).
The graph of the square root function and its degree 4 Taylor polynomial, centered at x=4.
The graph y=x is shown, for x in the interval [0,10]. Also shown is the degree 4 Taylor polynomial p4(x), centered at x=4. The Taylor polynomial appears to approximate x very well over the interval (2,7).
Figure 9.7.24. A graph of f(x)=x and its degree 4 Taylor polynomial at x=4
Our final example gives a brief introduction to using Taylor polynomials to solve differential equations.

Example 9.7.25. Approximating an unknown function.

A function y=f(x) is unknown save for the following two facts.
  1. y(0)=f(0)=1, and
  2. y=y2
(This second fact says that amazingly, the derivative of the function is actually the function squared!)
Find the degree 3 Maclaurin polynomial p3(x) of y=f(x).
Solution.
One might initially think that not enough information is given to find p3(x). However, note how the second fact above actually lets us know what y(0) is:
y=y2y(0)=y2(0).
Since y(0)=1, we conclude that y(0)=1.
Now we find information about y. Starting with y=y2, take derivatives of both sides, with respect to x. That means we must use implicit differentiation.
y=y2ddx(y)=ddx(y2)y=2yy.
Now evaluate both sides at x=0:
y(0)=2y(0)y(0)y(0)=2.
We repeat this once more to find y(0). We again use implicit differentiation; this time the Product Rule is also required.
ddx(y)=ddx(2yy)y=2yy+2yy.
Now evaluate both sides at x=0:
y(0)=2y(0)2+2y(0)y(0)y(0)=2+4=6.
In summary, we have:
y(0)=1y(0)=1y(0)=2y(0)=6.
We can now form p3(x):
p3(x)=1+x+22!x2+63!x3=1+x+x2+x3.
It turns out that the differential equation we started with, y=y2, where y(0)=1, can be solved without too much difficulty:
y=11x.
Figure 9.7.26 shows this function plotted with p3(x). Note how similar they are near x=0.
The graph of the exact solution to the differential equation in this example, and a polynomial approximation.
The graph y=1/(1x) is shown, for x from 1 to about 0.6. Also shown is the graph of p3(x), the Maclaurin polynomial obtained as an approximate solution to the differential equation in this example. As expected, the polynomial is a good approximation to the exact solution when x is close to 0.
Figure 9.7.26. A graph of y=1/(x1) and y=p3(x) from Example 9.7.25
It is beyond the scope of this text to pursue error analysis when using Taylor polynomials to approximate solutions to differential equations. This topic is often broached in introductory Differential Equations courses and usually covered in depth in Numerical Analysis courses. Such an analysis is very important; one needs to know how good their approximation is. We explored this example simply to demonstrate the usefulness of Taylor polynomials.
Most of this chapter has been devoted to the study of infinite series. This section has taken a step back from this study, focusing instead on finite summation of terms. In the next section, we explore Taylor Series, where we represent a function with an infinite series.

Exercises Exercises

Terms and Concepts

1.
What is the difference between a Taylor polynomial and a Maclaurin polynomial?
2.
True or False? In general, pn(x) approximates f(x) better and better as n gets larger.
  • ?
  • True
  • False
3.
For some function f(x), the Maclaurin polynomial of degree 4 is p4(x)=6+3x4x2+5x37x4. What is p2(x)?
4.
For some function f(x), the Maclaurin polynomial of degree 4 is p4(x)=6+3x4x2+5x37x4. What is f(0)?

Problems

Exercise Group.
In the following exercises, find the Maclaurin polynomial of degree n for the given function.
5.
Find the Maclaurin polynomial of degree n=3 for f(x)=ex.
6.
Find the Maclaurin polynomial of degree n=8 for f(x)=sin(x).
7.
Find the Maclaurin polynomial of degree n=5 for f(x)=xex.
8.
Find the Maclaurin polynomial of degree n=6 for f(x)=tan(x).
9.
Find the Maclaurin polynomial of degree n=4 for f(x)=e2x.
10.
Find the Maclaurin polynomial of degree n=4 for f(x)=11x.
11.
Find the Maclaurin polynomial of degree n=4 for f(x)=11+x.
12.
Find the Maclaurin polynomial of degree n=7 for f(x)=11+x.
Exercise Group.
In the following exercises, find the Taylor polynomial of degree n, at x=c, for the given function.
13.
Find the Taylor polynomial for f(x)=x of degree n=4, at c=1.
14.
Find the degree n=4 Taylor polynomial for f(x)=ln(x+1), at c=1.
15.
Find the degree n=6 Taylor polynomial for f(x)=cos(x), at c=π/4.
16.
Find the degree n=5 Taylor polynomial for f(x)=sin(x), at c=π/6.
17.
Find the degree n=5 Taylor polynomial for f(x)=1x, at c=2.
18.
Find the degree n=8 Taylor polynomial for f(x)=1x2, at c=1.
19.
Find the degree n=3 Taylor polynomial for f(x)=1x2+1, at c=1.
20.
Find the degree n=2 Taylor polynomial for f(x)=x2cos(x), at c=π.
Exercise Group.
In the following exercises, approximate the function value with the indicated Taylor polynomial and give approximate bounds on the error.
21.
Approximate sin(0.1) with the Maclaurin polynomial of degree 3.
22.
Approximate cos(1) with the Maclaurin polynomial of degree 4.
23.
Approximate 10 with the Taylor polynomial of degree 2 centered at x=9.
24.
Approximate ln(1.5) with the Taylor polynomial of degree 3 centered at x=1.
Exercise Group.
The following exercises ask for an n to be found such that pn(x) approximates f(x) within a certain bound of accuracy.
25.
Find n such that the Maclaurin polynomial of degree n of f(x)=ex approximates e within 0.0001 of the actual value.
26.
Find n such that the Taylor polynomial of degree n of f(x)=x, centered at x=4, approximates 3 within 0.0001 of the actual value.
27.
Find n such that the Maclaurin polynomial of degree n of f(x)=cos(x) approximates cos(π/3) within 0.0001 of the actual value.
28.
Find n such that the Maclaurin polynomial of degree n of f(x)=sin(x) approximates cos(π) within 0.0001 of the actual value.
Exercise Group.
In the following exercises, find the nth term of the indicated Taylor polynomial.
29.
Find a formula for the nth term of the Maclaurin polynomial for f(x)=ex.
30.
Find a formula for the nth term of the Maclaurin polynomial for f(x)=cos(x).
31.
Find a formula for the nth term of the Maclaurin polynomial for f(x)=sin(x).
32.
Find a formula for the nth term of the Maclaurin polynomial for f(x)=11x.
33.
Find a formula for the nth term of the Maclaurin polynomial for f(x)=11+x.
34.
Find a formula for the nth term of the Taylor polynomial for f(x)=ln(x) centered at x=1.
Exercise Group.
In the following exercises, approximate the solution to the given differential equation with a degree 4 Maclaurin polynomial.
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