10.3.2Can I improve the approximation?

For the function $f(x)=\cos(x)$:

1. Sketch the graph of $y = f(x)$ over the interval $- \frac { \pi } { 2 } \leq x \leq \frac { \pi } { 2 }$. On the same graph, sketch the line tangent to the curve at the point $(0, 1)$.

2. Use your tangent line to approximate $\cos(0.3)$ and compare it with $f(0.3)$. What is the difference?

3. Now sketch the curve $p ( x ) = 1 - \frac { 1 } { 2 } x ^ { 2 }$ on the same set of axes. What point is common to all three functions? Compare $p(0.3)$ with $f(0.3)$. What is the difference?

Since power series are functions of $x$, the graph of a power series may be useful.

1. Sketch the function $p ( x ) = 1 - \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 4 } } { 4 ! } - \frac { x ^ { 6 } } { 6 ! } + \frac { x ^ { 8 } } { 8 ! }$ for $–4 ≤ x ≤ 4$ and $–2 ≤ y ≤ 2$. What function does $p\left(x\right)$ resemble? Test your theory by simultaneously graphing the other function.

2. Does $p(x)$ give exact values of the function it resembles? Justify your answer.

3. Now sketch the function for $–6 ≤ x ≤ 4$. Write down your observations.

4. The function in part (a) is an eighth-degree degree polynomial. To get a more precise approximation, extend the series so that the function is a tenth-degree polynomial. Write the tenth-degree polynomial both in sigma notation and expanded form.

5. How can you alter the series expression from part (d) to make $p(x)$ an even better approximation of the function it resembles?

How can we determine “how good” an approximation function is? Consider $f(x) =\sqrt { x }$.

1. Write the equation of the line tangent to $f(x) =\sqrt { x }$ at $x = 1$.

2. Graph both $y = f(x)$ and the tangent line at $x = 1$ for $0 ≤ x ≤ 4$ and $0 ≤ y ≤ 3$. Then calculate the error of the tangent line at $x = 0$ and $x = 2$.

3. One drawback to using tangent lines for approximations is that they do not curve to follow the function. To use an approximation function which curves, create a quadratic polynomial $p(x) = ax^2 + bx + c$ so that it not only shares the point and the slope, but also has the same second derivative of $f$ at $x = 1$.

4. Add your quadratic function to your sketch from part (b). Calculate the error of the quadratic function at $x = 0$ and $x = 2$.

Determine the radius and interval of convergence for each of the following power series. Homework Help ✎

Write the equation of a quadratic function $p(x)=ax^2+bx+c$ which best approximates $f(x) = e^x$ about the point $(0, 1)$. Then calculate the error if the quadratic function is used to approximate $e^{0.7}$. Homework Help ✎

For the differential equation $\frac { d y } { d x } = 2 y ( 1 - y )$: Homework Help ✎

1. What type of growth is described by this equation? Finish the sentence, “The rate of growth is proportional to…”

2. Solve the differential equation with initial condition $y(0) = 0.5$ and sketch the solution curve.

Region $R$ is bounded by $f(x) = ax^2$, the $x$-axis, and the line $x = \frac { 1 } { a }$, where $a$ is a positive constant. 10-145 HW eTool (Desmos). Homework Help ✎

1. Sketch $R$ for three different values of $a$.

2. If $R$ is rotated about the $x$-axis, calculate the volume of the solid that is generated in terms of $a$.

Let $r_1 = 2 + \cos(θ)$, $r_2 = 2 + 2\cos(θ)$, and $r_3 = 2 + 3\cos(θ)$. 10-146 HW eTool (Desmos). Homework Help ✎

1. What are the minimum values of $r$ for each of the functions? What are the corresponding $θ$-values?

2. How does your answer to part (a) explain what happens to the graphs as $θ$ moves from $\frac { \pi } { 2 }$ to $\frac { 3 \pi } { 2 }$ ?

For the parametric equations $x(t) = 4t^2 - 4t$ and $y(t) = 1 - 4t^2$, write the equations of each horizontal tangent line to the curve and the corresponding $t$-value. Homework Help ✎