# Using calculus, it can be shown that the secant function can be approximated by the polynomial sec xapprox 1 + frac{x^{2}}{2!}+frac{5x^{4}}{4!} where x is in radians. Use a graphing utility to graph the secant function and its polynomial approximation in the same viewing window. How do the graphs compare?

Question
Polynomial graphs
Using calculus, it can be shown that the secant function can be approximated by the polynomial $$\displaystyle{\sec{{x}}}\approx{1}+{\frac{{{x}^{{{2}}}}}{{{2}!}}}+{\frac{{{5}{x}^{{{4}}}}}{{{4}!}}}$$ where x is in radians. Use a graphing utility to graph the secant function and its polynomial approximation in the same viewing window. How do the graphs compare?

2021-01-26

### Relevant Questions

Using calculus, it can be shown that the arctangent function can be approximated by the polynomial
$$\displaystyle{\arctan{\ }}{x}\ \approx\ {x}\ -\ {\frac{{{x}^{{{3}}}}}{{{3}}}}\ +\ {\frac{{{x}^{{{5}}}}}{{{5}}}}\ -\ {\frac{{{x}^{{{7}}}}}{{{7}}}}$$
a) Use a graphing utility to graph the arctangent function and its polynomial approximation in the same viewing window. How do the graphs compare?
b) Study the pattern in the polynomial approximation of the arctangent function and predict the next term. Then repeat part (a). How does the accuracy of the approximation change when an additional term is added?
In calculus, it can be shown that the arctangent function can be approximated by the polynomial
$$\displaystyle{\arctan{{x}}}\approx{x}-\frac{{x}^{{{3}}}}{{3}}+\frac{{x}^{{{5}}}}{{5}}-\frac{{x}^{{{7}}}}{{7}}$$
where x is in radians. Use a graphing utility to graph the arctangent function and its polynomial approximation in the same viewing window. How do the graphs compare?
Using calculus, it can be shown that the tangent function can be approximated by the polynomial $$\displaystyle{\tan{\ }}{x}\ \approx\ {x}\ +\ {\frac{{{2}{x}^{{{3}}}}}{{{3}!}}}\ +\ {\frac{{{16}{x}^{{{5}}}}}{{{5}!}}}$$ where x is in radians. Use a graphing utility to graph the tangent function and its polynomial approximation in the same viewing window. How do the graphs.
a) Identify the parameters a, k, d, and c in the polynomial function $$\displaystyle{y}={\frac{{{1}}}{{{3}}}}{\left[-{2}{\left({x}+{3}\right)}\right]}^{{{4}}}-{1}$$. Describe how each parameter transforms the base function $$\displaystyle{y}={x}^{{{4}}}$$. b) State the domain and range, the vertex, and the equation of the axis of symmetry of the transformed function. c) Describe two possible orders in which the transformations can be applied to the graph of $$\displaystyle{y}={x}^{{{4}}}$$ to produce the graph of $$\displaystyle{y}={\frac{{{1}}}{{{3}}}}{\left[-{2}{\left({x}+{3}\right)}\right]}^{{{4}}}-{1}$$. d) Sketch graphs of the base function and the transformed function on the same set of axes.
A 10 kg objectexperiences a horizontal force which causes it to accelerate at 5 $$\displaystyle\frac{{m}}{{s}^{{2}}}$$, moving it a distance of 20 m, horizontally.How much work is done by the force?
A ball is connected to a rope and swung around in uniform circular motion.The tension in the rope is measured at 10 N and the radius of thecircle is 1 m. How much work is done in one revolution around the circle?
A 10 kg weight issuspended in the air by a strong cable. How much work is done, perunit time, in suspending the weight?
A 5 kg block is moved up a 30 degree incline by a force of 50 N, parallel to the incline. The coefficient of kinetic friction between the block and the incline is .25. How much work is done by the 50 N force in moving the block a distance of 10 meters? What is the total workdone on the block over the same distance?
What is the kinetic energy of a 2 kg ball that travels a distance of 50 metersin 5 seconds?
A ball is thrown vertically with a velocity of 25 m/s. How high does it go? What is its velocity when it reaches a height of 25 m?
A ball with enough speed can complete a vertical loop. With what speed must the ballenter the loop to complete a 2 m loop? (Keep in mind that the velocity of the ball is not constant throughout the loop).
Investigate the change in the graph of a sinusoidal function of the form $$\displaystyle{\quad\text{and}\quad}={\sin{{x}}}{\quad\text{or}\quad}{\quad\text{and}\quad}={\cos{{x}}}$$ when multiplied by a polynomial function. Use a graphing calculator to sketch the graphs of and $$\displaystyle={2}{x},{\quad\text{and}\quad}=-{2}{x},{\quad\text{and}\quad}={2}{x}{\cos{{x}}}$$ on the same coordinate plane, on the interval $$\displaystyle{\left[-{20},{20}\right]}.$$
Rational functions can have any polynomial in the numerator and denominator. Analyse the key features of each function and sketch its graph. Describe the common features of the graphs. $$\displaystyle{a}{)}{f{{\left({x}\right)}}}={\frac{{{x}}}{{{x}^{{{2}}}-{1}}}}\ {b}{)}{g{{\left({x}\right)}}}={\frac{{{x}-{2}}}{{{x}^{{{2}}}+{3}{x}+{2}}}}\ {c}{)}{h}{\left({x}\right)}={\frac{{{x}+{5}}}{{{x}^{{{2}}}-{x}-{12}}}}$$
In the following items, you will analyze how several transformations affect the graph of the function $$\displaystyle{f{{\left({x}\right)}}}={\frac{{{1}}}{{{x}}}}$$. Investigate the graphs of $$\displaystyle{f{{\left({x}\right)}}}={\frac{{{1}}}{{{x}}}},{g{{\left({x}\right)}}}={f{{\left({x}\right)}}}={\frac{{{1}}}{{{x}+{2}}}},{h}{\left({x}\right)}={\frac{{{1}}}{{{x}-{2}}}},{p}{\left({x}\right)}={\frac{{{1}}}{{{x}-{4}}}}\ \text{and}\ {z}{\left({x}\right)}={\frac{{{1}}}{{{x}^{{{2}}}+{1}}}}$$. If you use a graphing calculator, select a viewing window $$\displaystyle\pm{23.5}$$ for x and $$\displaystyle\pm{15.5}$$ for y. At what values in the domain did vertical asymptotes occur for each of the functions? Explain why the vertical asymptotes occur at these values.
Determine whether $$F(x)=5x^{4}-\pi x^{3}+\frac{1}{2}$$ is a polynomial. If it is, state its degree. If not, say why it is not a polynomial. If it is a polynomial, write it in standard form. Identify the leading term and the constant term.
This exercise requires the use of a graphing calculator or computer programmed to do numerical integration. The normal distribution curve, which models the distributions of data in a wide range of applications, is given by the function $$p(x)=\frac{1}{\sqrt{2 \pi}^{\sigma}}e^{-(x-\mu)^{2}}/(2 \sigma^{2})$$ where $$\pi = 3.14159265 . . .$$ and sigma and mu are constants called the standard deviation and the mean, respectively. Its graph$$(\text{for}\ \sigma=1\ \text{and}\ \mu=2)$$is shown in the figure. With $$\sigma = 5 \text{and} \mu = 0$$, approximate $$\int_0^{+\infty}\ p(x)\ dx.$$