Polynomial Graphs: Examples and Samples

Find the exact length of the curve.
${\displaystyle x=e^t+e^{-t},y=5-2t,0\le t\le 3}$

Taylor’s formula with ${\displaystyle n=1}$ and ${\displaystyle a=0}$ gives the linearization of a function at ${\displaystyle x=0}$ With ${\displaystyle n=2}$ and ${\displaystyle n=3}$ we obtain the standard quadratic and cubic approximations. In these exercises we explore the errors associated with these approximations. We seek answers to two questions:
a) For what values of x can the function be replaced by each approximation with an error less than ${\displaystyle {10}^{-2}}$
b. What is the maximum error we could expect if we replace the function by each approximation over the specified interval? Using a CAS, perform the following steps to aid in answering questions (a) and (b) for the functions and interval.
Step 1: Plot the function over the specified interval.
Step 2: Find the Taylor polynomials ${\displaystyle P_1\left(x\right),P_2\left(x\right)\text{and}{P}_3\left(x\right)}$ at ${\displaystyle x=0}$
Step 3: Calculate the ${\displaystyle (n+1)}$ st derivative ${\displaystyle f^{(n+1)}\left(c\right)}$ associated with the remainder term for each Taylor polynomial. Plot the derivative as a function of c over the specified interval and estimate its maximum absolute value, M.
Step 4: Calculate the remainder ${\displaystyle R_n\left(x\right)}$ for each polynomial. Using the estimate M from Step 3 in place of ${\displaystyle f_{(n+1)}\left(c\right)}$ plot ${\displaystyle R_n\left(x\right)}$ over the specified interval. Then estimate the values of x that answer question (a).
Step 5: Compare your estimated error with the actual error ${\displaystyle E_n\left(x\right)=|f\left(x\right)-P_n\left(x\right)|}$ by plotting ${\displaystyle E_n\left(x\right)}$ over the specified interval. This will help answer question (b).
Step 6: Graph the function and its three Taylor approximations together. Discuss the graphs in relation to the information discovered in Steps 4 and 5.
${\displaystyle f\left(x\right)={(1+x)}^{\frac{3}{2}},-\frac{1}{2}\le x\le 2}$

Use Definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. ${\displaystyle f\left(x\right)=x^2+{(1+2x)}^{\frac{1}{2}},4\le x\le 7}$

Do you agree with the contention that the functions f(x) = x + 2 and g(x) = x 2 − 4 x – 2are the same in every respect. Provide evidence to support your position.

Previously, you have approximated curves with the graphs of Taylor polynomials. Discuss possible circumstances in which the osculating circle would be a better or worse approximation of a curve than the graph of a polynomial.

For the following exercises, use the given information about the polynomial graph to write the equation. Degree 5. Roots of multiplicity 2 at x = 3 and x = 1, and a root of multiplicity 1 at x = −3. y-intercept at (0, 9)

For the following exercises, use the given information about the polynomial graph to write the equation. Degree 3. Zeros at x = −5, x = −2, and x = 1. y-intercept at (0, 6)

For the following exercise, for each polynomial, a. find the degree. b. find the zeros, if any. c. find the y-intercept(s), if any. d. use the leading coefficient to determine the graph’s end behavior. and e. determine algebraically whether the polynomial is even, odd, or neither.
${\displaystyle f\left(x\right)=-3x^2+6x}$

For the following exercises, use the given information about the polynomial graph to write the equation. Degree 5. Roots of multiplicity 2 at x = −3 and x = 2 and a root of multiplicity 1 at x=−2. y-intercept at (0, 4).

For the following exercises, use the given information about the polynomial graph to write the equation. Degree 4. Root of multiplicity 2 at x = 4, and roots of multiplicity 1 at x = 1 and x = −2. y-intercept at (0, −3).

For the following exercise, for each polynomial, a. find the degree. b. find the zeros, if any. c. find the y-intercept(s), if any. d. use the leading coefficient to determine the graph’s end behavior. and e. determine algebraically whether the polynomial is even, odd, or neither.
${\displaystyle f\left(x\right)=x^3+3x^2-x-3}$

For the following exercises, use the given information about the polynomial graph to write the equation. Degree 5. Double zero at x = 1, and triple zero at x = 3. Passes through the point (2, 15).

If the result of a polynomial division is ${\displaystyle 2x^3-4x^2-5x+23x+2}$ , make at least three definitive statements

Do you agree with the contention that the functions $f(x)=x+2$ and $g(x)={\displaystyle \frac{x^2-4}{x-2}}$ are the same in every respect. Provide evidence to support your position.

Graph each polynomial function. Factor first if the expression is not in factored form. ${\displaystyle f\left(x\right)=2x(x-3)(x+2)}$

Graph each polynomial function. ${\displaystyle f\left(x\right)=2x^3+x^2+2x+1}$

Graph the polynomial by transforming an appropriate graph of the form ${\displaystyle y=x^n}$.
Show clearly all x- and y-intercepts. ${\displaystyle P\left(x\right)=-x^3+64}$

Copy and complete the anticipation guide in your notes. StatementThe quadratic formula can only be used when solving a quadratic equation.Cubic equations always have three real roots.The graph of a cubic function always passes through all four quadrants.The graphs of all polynomial functions must pass through at least two quadrants.The expression $x^2>4$ is only true if x>2.If you know the instantaneous rates of change for a function at x = 2 and x = 3, you can predict fairly well what the function looks like in between.
Agree Disagree Justification
Statement
The quadratic formula can only be used when solving a quadratic equation. Cubic equations always have three real roots. The graph of a cubic function always passes through all four quadrants. The graphs of all polynomial functions must pass through at least two quadrants.
The expression $x^2>4$ is only true if x>2. If you know the instantaneous rates of change for a function at x = 2 and x = 3, you can predict fairly well what the function looks like in between.

Graph each polynomial function. ${\displaystyle f\left(x\right)=4x^5-8x^4-x+2}$