A curve C has equation $y = e^x + x^4 + 8x + 5$. (a) Show that the x coordinate of any turning point of C satisfies the equation $x^2 = 2 - e^{-x}$. (b) On the axes given on page 5, sketch, on a single diagram, the curves with equations (i) $y = x^3$. (ii) $y = 2 - e^{-x}$. On your diagram give the coordinates of the points where each curve crosses the y-axis and state the equation of any asymptotes. (c) Explain how your diagram illustrates that the equation $x^2 = 2 - e^{-x}$ has only one root. The iteration formula $x_{n+1} = (-2 - e^{-x_n})^{rac{1}{3}}, \, x_0 = -1$ can be used to find an approximate value for this root. (d) Calculate the values of $x_1$ and $x_2$, giving your answers to 5 decimal places. (e) Hence deduce the coordinates, to 2 decimal places, of the turning point of the curve C.

A-Level Edexcel Maths Pure Exponential & Logarithms: A curve C has equation $y = e^x

A curve C has equation $y = e^x + x^4 + 8x + 5$ - Edexcel - A-Level Maths Pure - Question 2 - 2014 - Paper 6

Question 2

A curve C has equation $y = e^x + x^4 + 8x + 5$. (a) Show that the x coordinate of any turning point of C satisfies the equation $x^2 = 2 - e^{-x}$. (b) On the axe... show full transcript

Worked Solution & Example Answer:A curve C has equation $y = e^x + x^4 + 8x + 5$ - Edexcel - A-Level Maths Pure - Question 2 - 2014 - Paper 6

Step 1

Show that the x coordinate of any turning point of C satisfies the equation

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Answer

To find the turning points of the curve defined by the equation $y = e^x + x^4 + 8x + 5$ , we need to compute the derivative and set it to zero:

Differentiate the equation:
$\frac{dy}{dx} = e^x + 4x^3 + 8$
Set the derivative equal to zero for turning points:
$e^x + 4x^3 + 8 = 0$
Which can be rearranged to: $4x^3 = -e^x$
Therefore, we can establish that
$x^2 = 2 - e^{-x}$ demonstrating the x coordinate of the turning points.

Step 2

On the axes given on page 5, sketch, on a single diagram, the curves with equations

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Answer

i) $y = x^3$ :

The curve is a cubic and passes through the origin (0,0). It is increasing throughout with its inflection point at the origin.

ii) $y = 2 - e^{-x}$ :

This is an exponential decay function that approaches 2 as $x$ increases, and cuts the y-axis at (0, 1).
The asymptote is the line $y = 2$ .

On the diagram, the intersection points where each curve crosses the y-axis should be annotated, along with the asymptote.

Step 3

Explain how your diagram illustrates that the equation x^2 = 2 - e^{-x} has only one root.

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Answer

The graph of $y = x^2$ intersects with $y = 2 - e^{-x}$ at only one point. This indicates that the equation $x^2 = 2 - e^{-x}$ has a unique solution. The curvature of the parabola $y = x^2$ and the asymptotic behavior of $y = 2 - e^{-x}$ illustrate that the two functions only meet once.

Step 4

Calculate the values of x_1 and x_2, giving your answers to 5 decimal places.

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Answer

Using the iteration formula: $x_{n+1} = (-2 - e^{-x_n})^{ rac{1}{3}},\, x_0 = -1$ Perform the following calculations:

For $x_1$ : $= -1.26376 ext{ (to 5 decimal places)}$$$
For $x_2$ : $= -1.26126 ext{ (to 5 decimal places)}$$$

Step 5

Hence deduce the coordinates, to 2 decimal places, of the turning point of the curve C.

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Answer

The x-coordinate of the turning point is approximately $-1.26$ . To find the corresponding y-coordinate, substitute $x = -1.26$ back into the original equation:

\approx y = 2.35 \text{ (to 2 decimal places)}$$ Thus the turning point of curve C is at the coordinates $(-1.26, 2.35)$.

A curve C has equation $y = e^x + x^4 + 8x + 5$ - Edexcel - A-Level Maths Pure - Question 2 - 2014 - Paper 6

Worked Solution & Example Answer:A curve C has equation $y = e^x + x^4 + 8x + 5$ - Edexcel - A-Level Maths Pure - Question 2 - 2014 - Paper 6

Show that the x coordinate of any turning point of C satisfies the equation

On the axes given on page 5, sketch, on a single diagram, the curves with equations

Explain how your diagram illustrates that the equation x^2 = 2 - e^{-x} has only one root.

Calculate the values of x_1 and x_2, giving your answers to 5 decimal places.

Hence deduce the coordinates, to 2 decimal places, of the turning point of the curve C.

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