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Section Linear Approximations

Suppose we have a function \(f\) and a point \((p,f(p))\) on \(f.\) Suppose \(L\) is the tangent line to \(f\) at that point. \(L\) is the best linear approximation to \(f\) at \((p,f(p))\) since \(f(p) = L(p)\) and \(f'(p) = L'(p)\) and no line can do better than agreeing with \(f\) both in the y-coordinate and the derivative. Therefore the tangent line to \(f\) at \(p\) is often called the best linear approximation or the linearization of \(f\) at \(p.\)

Problem 3.1.

Find and sketch the linearization of the function at the indicated point.

  1. \(f(x)=3x^{2}-x+2\) at \(x=-2\text{.}\)

  2. \(g(x)=\sin(x)\) at \(x=\pi/4\text{.}\)

Problem 3.2.

Suppose \(f\) is a differentiable function and \((p,f(p))\) is a point on \(f.\) Find the equation of the tangent line to \(f\) at the point \((p,f(p)).\)

Problem 3.3.

Suppose \(f(x) = x^3.\) Find a parabola \(p(x) = ax^2 + bx + c\) so that \(f(1) = p(1)\) and \(f'(1) = p'(1)\) and \(f''(1) = p''(1).\) We would call \(p\) the best quadratic approximation to \(f\) at \((1,f(1)).\)

Example 3.4.

Approximating changes using linear approximations.

Suppose we have an ice cube floating in space that is expanding as cosmic dust forms on its exterior. Suppose the cube has an initial side length of \(s=3\) inches and we wish to approximate how much the volume (\(\dsp V=s^3\)) will change as the length of the side increases by \(.1\) inch. Of course, we can compute the change in volume directly. But we can also approximate it using the fact that the derivative tells us the rate of change of the volume. Recall that

\begin{equation*} V'(3) \approx \frac{V(3.1) - V(3)}{3.1 - 3}. \end{equation*}

If we think of the change in volume as \(\Delta V = V(3.1) - V(3)\) and the change in side length as \(\Delta s = 3.1 - 3\text{,}\) then

\begin{equation*} V'(3) \approx \frac{\Delta V}{\Delta s}. \end{equation*}

So,

\begin{equation*} \Delta V \approx V'(s) \Delta s = 3(3)^2 * .1 = 2.7 \end{equation*}

and we expect the volume to change by this amount. In a case where you knew the rate of change of the volume, but did not know a formula for volume you could use this to estimate the change in volume over a small change in side length.

Problem 3.5.

Compute the exact change in volume of the cube from the previous discussion and compare to our estimate.

Problem 3.6.

Suppose the radius of a circle \(C\) is to be increased from an initial value of 10 by an amount \(\Delta r = .01\text{.}\) Estimate the corresponding increase in the circle's area (\(\dsp A=\pi r^2\)) and compare it to the exact change.

Problem 3.7.

The surface area of a sphere (\(S=4\pi r^{2}\)) is expanding as air is pumped in. Approximate the change in the surface area as the radius expands from \(r=2.5\) to \(r=2.6\) and compare this to the exact change.

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