This TSA checks standard Maclaurin series, binomial expansion, range of validity, combining and manipulating series, Maclaurin series by differentiation, approximation and error, and small angle approximation.
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Standard Maclaurin Series and Formula Recognition
1. The Maclaurin series for \(e^x\), \(\sin x\) and \(\cos x\) are valid for all real \(x\).
Tested fundamental: Recognising the range of validity of standard series.
Explanation: The standard Maclaurin series for \(e^x\), \(\sin x\) and \(\cos x\) are valid for all real \(x\).
Common wrong choice: Choosing False usually comes from confusing these series with binomial or logarithmic series, which have restricted ranges of validity.
2. Which is the expansion of \(e^{-2x}\) up to and including the term in \(x^3\)?
Tested fundamental: Substituting into the standard series for \(e^x\).
Explanation: Use \(e^u=1+u+\frac{u^2}{2!}+\frac{u^3}{3!}+\cdots\) with \(u=-2x\). Hence \(e^{-2x}=1-2x+2x^2-\frac43x^3+\cdots\).
Common wrong choice: C treats every term after \(-2x\) as negative, but even powers of \(-2x\) are positive.
3. Find the coefficient of \(x^3\) in the expansion of \[ \ln(1+3x). \]
Answer format: Give your answer as an integer or exact fraction. Use an improper fraction if applicable.
Answer: \(9\)
Tested fundamental: Finding a coefficient from the standard series for \(\ln(1+x)\).
Explanation: \(\ln(1+3x)=3x-\frac{(3x)^2}{2}+\frac{(3x)^3}{3}+\cdots\), so the coefficient of \(x^3\) is \(\frac{27}{3}=9\).
Common error: Forgetting to cube the whole term \(3x\).
4. Which is the expansion of \(\sin 2x\) up to and including the term in \(x^3\)?
Tested fundamental: Substituting into the standard series for \(\sin x\).
Explanation: \(\sin 2x=2x-\frac{(2x)^3}{3!}+\cdots=2x-\frac43x^3+\cdots\).
Common wrong choice: A forgets that \((2x)^3=8x^3\).
Binomial Expansion and Range of Validity
5. Which is the expansion of \((1-2x)^{-1}\) up to and including the term in \(x^2\)?
Tested fundamental: Expanding \((1+u)^{-1}\) using the binomial series.
Explanation: \((1-2x)^{-1}=1+2x+(2x)^2+\cdots=1+2x+4x^2+\cdots\).
Common wrong choice: B treats the \(x\)-term as negative instead of recognising the geometric-type pattern.
6. For non-integer \(n\), find the range of validity of the binomial expansion of \[ (1+3x)^n. \]
Answer format: Give your answer in interval notation. For example [0,1) or (-inf,5].
Answer: \(\displaystyle \left(-\frac13,\frac13\right)\)
Tested fundamental: Finding the range of validity for a binomial expansion.
Explanation: For \((1+u)^n\), the binomial series is valid when \(|u|<1\). Here \(u=3x\), so \(|3x|<1\). Hence \(-\frac13<x<\frac13\).
Common error: Forgetting that the condition is on the expression replacing \(u\), not just on \(x\) itself.
7. Which is the expansion of \((x+2)^{-1}\) in descending powers of \(x\), up to and including the term in \(x^{-3}\)?
Tested fundamental: Rewriting a binomial expression for expansion in descending powers of \(x\).
Explanation: \((x+2)^{-1}=x^{-1}\left(1+\frac2x\right)^{-1}\). Then \(\left(1+\frac2x\right)^{-1}=1-\frac2x+\frac4{x^2}+\cdots\), so \((x+2)^{-1}=x^{-1}-2x^{-2}+4x^{-3}+\cdots\).
Common wrong choice: A misses the alternating signs from \((1+u)^{-1}\).
8. The expansion of \(\ln(1-2x)\) is valid for \[ -\frac12<x\le \frac12. \]
Tested fundamental: Range of validity for logarithmic series.
Explanation: For \(\ln(1+u)\), the range is \(-1<u\le1\). Here \(u=-2x\), so \(-1<-2x\le1\). Dividing by \(-2\) reverses the inequalities, giving \(-\frac12\le x<\frac12\).
Common wrong choice: Choosing True usually comes from not reversing the inequality signs after dividing by a negative number.
Combining, Multiplying and Dividing Series
9. Which is the expansion of \[ \cos x+e^{-x} \] up to and including the term in \(x^2\)?
Tested fundamental: Adding two standard series and simplifying like terms.
Explanation: \(\cos x=1-\frac{x^2}{2}+\cdots\) and \(e^{-x}=1-x+\frac{x^2}{2}+\cdots\). Adding gives \(\cos x+e^{-x}=2-x+0x^2+\cdots\).
Common wrong choice: B fails to combine the \(x^2\)-terms correctly.
10. To expand \[ \frac{\ln(1+x)}{4-x^2} \] up to and including the term in \(x^3\), which rewrite of the denominator factor is most useful?
Tested fundamental: Rewriting division of series as a product of series.
Explanation: Pull out the constant \(4\): \(4-x^2=4\left(1-\frac{x^2}{4}\right)\). Therefore \(\frac{1}{4-x^2}=\frac14\left(1-\frac{x^2}{4}\right)^{-1}\).
Common wrong choice: A factors the expression incorrectly because \(4\left(1-\frac{x}{2}\right)=4-2x\), not \(4-x^2\).
11. When expanding a series within a series, such as \[ \cos(2\sin x), \] which step is usually most appropriate first?
Tested fundamental: Handling a series within a series.
Explanation: For a series within a series, work from the inside out. First expand the inner expression, then substitute it into the outer standard series.
Common wrong choice: B often leads to unnecessary or incorrect manipulation because the inner function has not yet been simplified.
Maclaurin Series by Differentiation and Deducing New Series
12. In the Maclaurin expansion of \(\cos x\), which value explains why there is no \(x\)-term?
Tested fundamental: Connecting Maclaurin coefficients to derivatives at \(x=0\).
Explanation: The coefficient of \(x\) is \(f'(0)\). For \(f(x)=\cos x\), \(f'(x)=-\sin x\), so \(f'(0)=0\). Hence there is no \(x\)-term.
Common wrong choice: B is about the constant term, not the \(x\)-term.
13. If \[ f(0)=2,\qquad f'(0)=-1,\qquad f”(0)=6, \] which is the Maclaurin expansion of \(f(x)\) up to and including the term in \(x^2\)?
Tested fundamental: Building a Maclaurin expansion from derivative values.
Explanation: \(f(x)=f(0)+xf'(0)+\frac{x^2}{2!}f”(0)+\cdots\). So \(f(x)=2-x+\frac{6}{2}x^2+\cdots=2-x+3x^2+\cdots\).
Common wrong choice: A forgets the factorial divisor \(2!\) in the \(x^2\)-term.
14. Why can the Maclaurin series of \(\cot x\) not be found about \(x=0\)?
Tested fundamental: Conditions for forming a Maclaurin series.
Explanation: A Maclaurin series is centred at \(x=0\), so the function must be defined at \(x=0\). But \(\cot0\) is undefined.
Common wrong choice: D is irrelevant; many trigonometric functions do have Maclaurin series.
15. Given the Maclaurin series for \(f(x)\), which is the most appropriate way to obtain the series for \[ \frac{f'(x)}{f(x)} \] up to a required order?
Tested fundamental: Deducing a related series from a known series.
Explanation: First differentiate the series term-by-term to get \(f'(x)\), then divide by the original series for \(f(x)\), keeping terms up to the required order.
Common wrong choice: D only gives information at \(x=0\) and does not produce a series.
Approximation, Error and Validity
16. Using \[ \frac{1}{3+\sin2x} \approx \frac13-\frac29x+\frac{4}{27}x^2, \] estimate \[ \int_0^{1.5}\frac{1}{3+\sin2x}\,dx. \]
Answer format: Give your answer as an exact fraction. Use an improper fraction if applicable.
Answer: \(\displaystyle \frac5{12}\)
Tested fundamental: Using a Maclaurin approximation inside a definite integral.
Explanation: Integrate the approximate polynomial: \(\int_0^{1.5}\left(\frac13-\frac29x+\frac{4}{27}x^2\right)dx=\left[\frac{x}{3}-\frac{x^2}{9}+\frac{4x^3}{81}\right]_0^{1.5}\). Since \(1.5=\frac32\), this gives \(\frac12-\frac14+\frac16=\frac5{12}\).
Common error: Trying to integrate the original function instead of using the given approximation.
17. Adding more terms to a Maclaurin expansion always improves the approximation, even outside the range of validity of the series.
Tested fundamental: Understanding approximation reliability and validity.
Explanation: More terms generally improve the approximation only when the series is being used within its range of validity. Outside the range of validity, the series may not converge to the function.
Common wrong choice: Choosing True ignores the importance of the range of validity.
18. Which formula gives the percentage error when \(F(x)\) is used to approximate \(f(x)\)?
Tested fundamental: Percentage error formula for approximations.
Explanation: Percentage error is \(\left|\frac{\text{actual value}-\text{approximate value}}{\text{actual value}}\right|\times100\%\). Here \(f(x)\) is the actual value and \(F(x)\) is the approximation, so the denominator should be \(f(x)\).
Common wrong choice: B divides by the approximate value \(F(x)\), not the actual value.
Small Angle Approximation and Applications
19. For sufficiently small \(x\), which set of small-angle approximations is appropriate up to and including the lowest useful non-zero terms?
Tested fundamental: Recalling standard small-angle approximations.
Explanation: For small \(x\), \(\sin x\approx x\), \(\cos x\approx 1-\frac{x^2}{2}\), and \(\tan x\approx x\).
Common wrong choice: B incorrectly treats \(\sin x\) as close to \(1\), but \(\sin0=0\).
20. For sufficiently small \(\theta\), \[ \frac{1}{\cos\theta+\sqrt3\sin\theta} \approx 1-\sqrt3\theta+k\theta^2. \] Find \(k\).
Answer format: Give your answer as an exact fraction. Use an improper fraction if applicable.
Answer: \(\displaystyle \frac72\)
Tested fundamental: Applying small-angle approximation and binomial expansion.
Explanation: Use \(\cos\theta\approx1-\frac{\theta^2}{2}\) and \(\sin\theta\approx\theta\). Then \(\cos\theta+\sqrt3\sin\theta\approx1+\sqrt3\theta-\frac{\theta^2}{2}\). Expanding the reciprocal gives \(1-\sqrt3\theta+\frac{\theta^2}{2}+3\theta^2=1-\sqrt3\theta+\frac72\theta^2\). Hence \(k=\frac72\).
Common error: Forgetting the \(u^2\) term when expanding \((1+u)^{-1}\).
21. In a triangle small-angle approximation problem, which sequence is usually most appropriate?
Tested fundamental: Method structure for triangle small-angle approximation problems.
Explanation: First obtain a correct expression from the geometry, then apply small-angle approximation and series expansion.
Common wrong choice: A applies approximation too early and may ignore fixed angles or the correct trigonometric relationship.
22. Which statement about small-angle approximation is most accurate?
Tested fundamental: Understanding when small-angle approximation can be applied.
Explanation: Multiples like \(2x\) and \(3x\) are still small when \(x\) is small. But \(x+a\) is not necessarily small when \(a\) is a non-zero constant.
Common wrong choice: D ignores that adding a constant changes the angle being approximated.
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