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Chapter 23: Problem 652

Find the equation of the curve which has a horizontal tangent at the point \((0,-1)\), and for which the rate of change, with respect to \(\mathrm{x}\), of the slope at any point is equal to \(8 \mathrm{e}^{2 \mathrm{x}}\).

Short Answer

Expert verified
The equation of the curve is \(f(x) = 2e^{2x} - 4x - 3\).

Step by step solution

01

Identify the given rate of change of the slope

The rate of change of the slope (i.e., the second derivative) with respect to \(x\) is given by \(8e^{2x}\).
02

Integrate to find the slope (first derivative)

To find the slope (first derivative) of the curve, we integrate the second derivative, which we denote by \(f''(x)\): \(f''(x) = 8e^{2x}\) Now integrate with respect to \(x\): \(f'(x) = \int 8e^{2x} dx = 4e^{2x} + C_1\) Where \(C_1\) is the integration constant.
03

Use the information about the horizontal tangent to find \(C_1\)

The curve has a horizontal tangent (slope = 0) at the point \((0, -1)\). This information can be used to find the value of \(C_1\). So, plug in the point \((0, -1)\) to the expression of the first derivative: \(f'(0) = 4e^{2\cdot0} + C_1 = 0\) Therefore, \(C_1 = -4\) and the expression for the first derivative (slope) becomes: \(f'(x) = 4e^{2x} - 4\)
04

Integrate to find the curve's equation (the function itself)

Now, integrate the first derivative to find the equation of the curve, which we denote by \(f(x)\): \(f(x) = \int (4e^{2x} - 4) dx = 2e^{2x} - 4x + C_2\) Where \(C_2\) is the integration constant.
05

Use the given point to find \(C_2\)

We know that the curve passes through the point \((0, -1)\). This information can be used to find the value of \(C_2\). Plug in the point \((0, -1)\) to the expression of the function: \(f(0) = 2e^{2\cdot0} - 4\cdot0 + C_2 = -1\) Therefore, \(C_2 = -3\) and the expression for the curve's equation becomes: \(f(x) = 2e^{2x} - 4x - 3\) Thus, the equation of the curve is:
06

Final Answer

\(f(x) = 2e^{2x} - 4x - 3\)

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Most popular questions from this chapter

Integrate: \(\int\left[(\mathrm{x}+2) /\left(\mathrm{x}^{2}+16\right)\right] \mathrm{dx}\)

Integrate: \(\int[2 \mathrm{x} /(\mathrm{x}+1)] \mathrm{dx}\)

\((\mathrm{dy} / \mathrm{dx})=\left[2 /\left(\varepsilon^{2 \mathrm{x}}+\varepsilon^{-2 \mathrm{x}}\right)\right] .\) Find \(\mathrm{y}=\mathrm{F}(\mathrm{x})\)

Integrate the expression: \(\left.\int \sqrt{(} 16-9 \mathrm{x}^{2}\right) \mathrm{d} \mathrm{x}\).

$$ \begin{aligned} &(\mathrm{dy} / \mathrm{dx})=(\mathrm{a}+\mathrm{bx})^{\mathrm{n}} . \text { What is } \mathrm{y}=\mathrm{F}(\mathrm{x}) \text { when } \mathrm{n}=1, \mathrm{n}=2, \text { and }\\\ &\mathrm{n}=-2 ? \end{aligned} $$
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