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Derivatives – test 5: Taylor and Maclaurin formula

10 multiple-choice and short-answer problems on derivatives (Mathematics I). The solutions are shown after checking.

For multiple-choice questions choose one answer. For the others type the answer: a number, a decimal or a fraction such as 5/12. When you finish, click Check answers.

1. \(P(x)=x^4+2x^3-3x^2+4x+1\) is expanded in powers of \(x-2\). What is the constant term?Enter a number only

2. \(P(x)=x^4+2x^3-3x^2+4x+1\) is expanded in powers of \(x-2\). What is the coefficient of \((x-2)^2\)?Enter a number only

3. \(P(x)=x^3-2x+5\) is expanded in powers of \(x-1\). What is the coefficient of \((x-1)\)?Enter a number only

4. What is \(P^{V}(0)\) for \(P(x)=2x^6-10x^5+7x^3-5x^2+4x+3\)?Enter a number only

5. What is the coefficient of \(x^4\) in the Maclaurin expansion of \(e^x\)?Enter a number only

6. Which is the \(x^4\) term in the Maclaurin expansion of \(\cos x\)?

A. \(-\frac{x^4}{24}\)

B. \(\frac{x^4}{4}\)

C. \(-\frac{x^4}{12}\)

D. \(\frac{x^4}{24}\)

7. What is the coefficient of \(x^3\) in the Maclaurin expansion of \(\ln(1+x)\)?Enter a number only

8. Approximate \(\cos0.2\) with the Maclaurin expansion up to the \(x^4\) term (to 7 decimal places).Enter a number only

9. In the Taylor expansion of \(f(x)=x\arctanx\) about \(x_0=1\), what is the coefficient of \((x-1)^2\)?Enter a number only

10. Approximate \(\sqrt e\) with the expansion of \(e^x\) up to the \(x^3\) term (to 4 decimal places).Enter a number only