4.08a Maclaurin series: find series for function

The function \(f\) is given by $$f(x) = e^x \cos x.$$

1. Show that \(f''(x) = -2e^x \sin x\). [2]
2. Determine the Maclaurin series for \(f(x)\) as far as the \(x^4\) term. [6]
3. Hence, by differentiating your series, determine the Maclaurin series for \(e^x \sin x\) as far as the \(x^3\) term. [4]
4. The equation $$10e^x \sin x - 11x = 0$$ has a small positive root. Determine its approximate value, giving your answer correct to three decimal places. [4]

In this question you must show all stages of your working. The function \(f\) is defined by \(f(x) = (1 + 2x)^{\frac{1}{2}}\).

1. Find \(f'''(x)\) (i.e. the third derivative of \(f\)) showing all your intermediate steps. Hence, find the Maclaurin series for \(f(x)\) up to and including the \(x^3\) term. [8 marks]
2. Use the expansion of \(e^x\) together with the result in part (a) to show that, up to and including the \(x^3\) term, $$e^x(1 + 2x)^{\frac{1}{2}} = 1 + 2x + x^2 + kx^3,$$ where \(k\) is a rational number to be found. [3 marks]

Let $$f(x) = \frac{27x^2 + 32x + 16}{(3x + 2)^2(1 - x)}$$

1. Express \(f(x)\) in terms of partial fractions [5]
2. Hence, or otherwise, find the series expansion of \(f(x)\), in ascending powers of \(x\), up to and including the term in \(x^2\). Simplify each term. [6]
3. State, with a reason, whether your series expansion in part (b) is valid for \(x = \frac{1}{2}\). [2]

\(y = \cosh^n x\) \quad \(n \geq 5\)

1. 1. Show that $$\frac{d^2y}{dx^2} = n^2\cosh^n x - n(n-1)\cosh^{n-2}x$$ [4]
   2. Determine an expression for \(\frac{d^4y}{dx^4}\) [2]
2. Hence, or otherwise, determine the first three non-zero terms of the Maclaurin series for \(y\), simplifying each coefficient and justifying your answer. [2]

Given that $$y = \cos x \sinh x \quad x \in \mathbb{R}$$

1. show that $$\frac{d^4y}{dx^4} = ky$$ where \(k\) is a constant to be determined. [5]
2. Hence determine the first three non-zero terms of the Maclaurin series for \(y\), giving each coefficient in simplest form. [3]

You are given that \(f(x) = \ln(2 + x)\).

1. Determine the exact value of \(f'(0)\). [2]
2. Show that \(f''(0) = -\frac{1}{4}\). [2]
3. Hence write down the first three terms of the Maclaurin series for \(f(x)\). [3]

The Taylor series expansion, about \(x = 1\), of the function \(y\) is $$y = 1 + \sum_{n=1}^{\infty} \frac{(-2)^{n-1}(x-1)^n}{1 \times 3 \times 5 \times \ldots \times (2n-1)}.$$ Find the value of \(\frac{\text{d}^4 y}{\text{d}x^4}\) when \(x = 1\). [3]

The function \(y\) satisfies \(\frac{d^2y}{dx^2} + x^2y = x\), and is such that \(y = 1\) and \(\frac{dy}{dx} = 1\) when \(x = 1\).

1. Using the given differential equation
   1. state the value of \(\frac{d^2y}{dx^2}\) when \(x = 1\), [1]
   2. find, by differentiation, the value of \(\frac{d^3y}{dx^3}\) when \(x = 1\). [2]
2. Hence determine the Taylor series for \(y\) about \(x = 1\) up to and including the term in \((x-1)^3\) and deduce, correct to 4 decimal places, an approximation for \(y\) when \(x = 1.1\). [3]

Given that \(y = \cos\{\ln(1 + x)\}\), prove that

1. \((1 + x)\frac{\mathrm{d}y}{\mathrm{d}x} = -\sin\{\ln(1 + x)\}\), [1]
2. \((1 + x)^2 \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (1 + x)\frac{\mathrm{d}y}{\mathrm{d}x} + y = 0\). [2]

Obtain an equation relating \(\frac{\mathrm{d}^3 y}{\mathrm{d}x^3}\), \(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}\) and \(\frac{\mathrm{d}y}{\mathrm{d}x}\). [2] Hence find Maclaurin's series for \(y\), up to and including the term in \(x^3\). [4] Verify that the same result is obtained if the standard series expansions for \(\ln(1 + x)\) and \(\cos x\) are used. [3]