4.5 An Algorithm for Curve Sketching
Chapter
Chapter 4
Section
4.5
Solutions 23 Videos

If a polynomial function of degree three has a local minimum, explain how the functions values behave as x\to \infty and x\to -\infty. Consider all cases.

1.19mins
Q1

How many local maximum and local minimum values are possible for a polynomial function of degree three, four, or n? Explain.

0.47mins
Q2

Determine whether each function has vertical asymptotes. If it does, state the equations of the asymptotes.

y = \displaystyle{\frac{x}{x^2 +4x + 3}}

0.19mins
Q3a

Determine whether each function has vertical asymptotes. If it does, state the equations of the asymptotes.

y = \displaystyle{\frac{5x -4}{x^2-6x + 12}}

0.42mins
Q3b

Determine whether each function has vertical asymptotes. If it does, state the equations of the asymptotes.

y = \displaystyle{\frac{3x + 2}{x^2-6x + 9}}

0.17mins
Q3c

Use the algorithm for curve sketching to sketch the following:

\displaystyle y = x^3 - 9x^2 + 15x +30

3.54mins
Q4a

Use the algorithm for curve sketching to sketch the following:

\displaystyle f(x) = -4x^3 + 18x^2 + 3

2.08mins
Q4b

Use the algorithm for curve sketching to sketch the following:

\displaystyle y = 3 + \frac{1 }{(x + 2)^2}

2.47mins
Q4c

Use the algorithm for curve sketching to sketch the following:

\displaystyle f(x) = x^4 -4x^3 - 9x^2 + 48x

7.26mins
Q4d

Use the algorithm for curve sketching to sketch the following:

\displaystyle y = \frac{2x}{x^2 - 25}

5.32mins
Q4e

Use the algorithm for curve sketching to sketch the following:

\displaystyle f(x) =\frac{1}{x^2 -4x}

7.12mins
Q4f

Use the algorithm for curve sketching to sketch the following:

\displaystyle y = \frac{6x^2 -2}{x^3}

6.00mins
Q4g

Use the algorithm for curve sketching to sketch the following:

\displaystyle y = \frac{x + 3}{x^2 -4}

8.49mins
Q4h

Use the algorithm for curve sketching to sketch the following:

\displaystyle y = \frac{x^2 -3x +6}{x -1}

5.43mins
Q4i

Use the algorithm for curve sketching to sketch the following:

\displaystyle y = (x - 4)^{\displaystyle{\frac{2}{3}}}

2.18mins
Q4j

Determine the constants a, b, c, and d so that the curve defined by y = ax^3 + bx^2 +cx +d has a local maximum at the point (2, 4) and a point of inflection at the origin.

4.44mins
Q6

Given the following results of the analysis of a function, sketch a possible graph for the function:

f(0) = 0, the horizontal asymptote is y =2, the vertical asymptote is x = 3 and f'(x) < 0 and f''(x) < 0 for x < 3 and f''(x) > 0 for x > 3.

1.25mins
Q7a

Give the following results of the analysis of a function, sketch a possible graph for the function:

f(0) = 6, f(-2) = 0 the horizontal asymptote is y = 7, the vertical asymptote is x = -4, and f'(x) > 0 and f''(x) > 0 for x < -4; f'(x) > 0 and f''(x)< 0 for x > -4.

3.28mins
Q7b

Sketch the graph of f(x) = \displaystyle{\frac{k -x}{k^2 + x^2}}, where k is any positive constant.

14.12mins
Q8

Sketch the cute defined by g(x) = x^{{\frac{1}{3}}}(x + 3)^{{\frac{2}{3}}}.

9.29mins
Q9

Find the horizontal asymptotes.

f(x) = \displaystyle{\frac{x}{\sqrt{x^2 + 1}}}

1.21mins
Q10a

Find the horizontal asymptotes.

g(t) = \sqrt{t^2+4t} - \sqrt{t^2 + t}

3.02mins
Q10b

Show that, for any cubic function of the form y = ax^3 + bx^2 + cx + d, there is a single point of inflection, and the slope of the cute at that point is c-\displaystyle{\frac{b^2}{3a}}.

1.39mins
Q11
Lectures 2 Videos

Full Graphing ex1

Sketch f(x) = x^4 -8x^3

10.22mins
Full Graphing ex1

Full Graphing ex2

Sketch \displaystyle f(x) =x^{\frac{2}{3}}(6-x)^{\frac{1}{3}}