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Using basic derivatives to calculate the gradient function of a hill $y=-e^{x}+b\\ln{\\left(x\\right)+c$, and then substituting values to find the gradient at specific distance from the sea.

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The cross-section of a hill with a steep cliff face is modelled by the equation:
\\[y=-e^{x}+\\var{b}\\ln{\\left(x\\right)+\\var{c}}\\]
where $y$ is the height of the ground above sea level, measured in decametres ($\\mathrm{dam}$), and $x$ is the horizontal distance from the sea, also measured in decametres.

The model applies for values of $x$ in the range $0.01\\le\\ x\\le3$.

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a) To find the gradient of the hill at any given horizontal distance from the sea, $x$, we need to differentiate the equation

\\[y=-e^{x}+\\var{b}\\ln{\\left(x\\right)+\\var{c}}\\]

with respect to x:

\\[ \\begin{split} \\frac{dy}{dx}&=-e^x+\\var{b}\\times\\frac{1}{x}+0 \\\\&=-e^x+\\frac{\\var{b}}{x} \\end{split}\\]

\\[ \\begin{split} \\frac{dy}{dx}&=-e^x+\\frac{1}{x}+0 \\\\&=-e^x+\\frac{\\var{b}}{x} \\end{split}\\]

Therefore, the formula that describes the gradient of the hill at any horizontal distance from the sea is:

\\[ \\begin{split} \\frac{dy}{dx}&=-e^x+\\frac{\\var{b}}{x} \\end{split}\\]

b) We can use the formula from part (a) to calculate the gradient at specific points.

Remember that the distance in this problem is measured in decametres ($\\mathrm{dam}$) and $1 ~~\\mathrm{dam}$ = $10~~ \\mathrm{m}$.

So, when $x=1$:

\\[ \\begin{split} \\frac{dy}{dx}&=-e^1+\\frac{\\var{b}}{1} \\\\&=-e+\\var{b} \\\\ &= \\var{ansb1} ~ \\text{[2 d.p.]}\\end{split}\\]

Thus, the gradient of the hill at a horizontal distance of $1~~\\mathrm{dam}$ from the sea is $\\frac{dy}{dx}=\\var{ansb1} ~~ \\mathrm{dam}$.

When $x=0.1$:

\\[ \\begin{split} \\frac{dy}{dx}&=-e^{0.1}+\\frac{\\var{b}}{0.1} \\\\ &= \\var{ansb2} ~ \\text{[2 d.p.]}\\end{split}\\]

Thus, the gradient of the hill at a horizontal distance of $0.1~~\\mathrm{dam}$ from the sea is $\\frac{dy}{dx}=\\var{ansb2} ~~ \\mathrm{dam}$.

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Find the formula that describes the gradient of the hill at any given horizontal distance from the sea.

$\\frac{dy}{dx}=$ [[0]].

What is the gradient of the hill at a horizontal distance of:

$10$ metres ($1 ~~\\mathrm{dam}$) from the sea?

[[0]] $\\mathrm{dam}$

$1$ metre ($0.1~~ \\mathrm{dam}$) from the sea?

[[1]] $\\mathrm{dam}$

Give your answers to 2 decimal places when necessary.

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