Problem 65 - Calculus Early Transcendentals

Question

More graphing Sketch a complete graph of the following functions. Use analytical methods and a graphing utility together in a complementary way. $$f(x)=10 x^{6}-36 x^{5}-75 x^{4}+300 x^{3}+120 x^{2}-720 x$$

Step-by-Step Solution

Answer

Question: Sketch a complete graph of the function $$f(x)=10x^6-36x^5-75x^4+300x^3+120x^2-720x$$, and identify its key features. Answer: To sketch a complete graph of the given function, follow these steps: 1. Find x-intercepts: Set the function equal to 0 and solve for x. The x-intercepts are approximately at x = 0.4308, x = 1.5859, x = 2.9569, x = 3.2814, and x = 4.745. 2. Find y-intercept: Set x equal to 0 and solve for y. The y-intercept is at the point (0,0). 3. Find critical points: Compute the derivative and find where the derivative equals 0 or is undefined. The critical points are approximately at x = 0.5586, x = 1.9713, and x = 4.0653. 4. Find inflection points: Compute the second derivative and find where the second derivative changes sign. The inflection points are approximately at x = 0.7404, x = 1.6869, x = 3.9821, and x = 4.5903. 5. Piece it all together in a graph: Use a graphing utility to plot the function and verify these findings. Adjust the sketch as necessary based on the graphing utility's output. Using this information, we can sketch a complete graph of the function and identify its key features.

Step 1: Find x-intercepts

To find the x-intercepts, set the function equal to 0: $$10 x^6-36 x^5-75 x^4+300 x^3+120 x^2-720 x = 0$$ Notice that every term has an x in it, so we can factor out an x: $$x(10 x^5-36 x^4-75 x^3+300 x^2+120 x-720) = 0$$ Now, we can use a graphing calculator or other tools to find the roots of the remaining polynomial and the x-intercepts.

Step 2: Find y-intercept

To find the y-intercept, set x equal to 0 and solve for y: $$f(0) = 10(0)^6-36(0)^5-75(0)^4+300(0)^3+120(0)^2-720(0) = 0$$ So the y-intercept is at the point (0,0).

Step 3: Find critical points

To find critical points (i.e., where the derivative equals 0 or is undefined), first compute the derivative: $$f'(x) = 60x^5 - 180x^4 - 300x^3 + 900x^2 + 240x - 720$$ Now look for the points where $$f'(x)=0$$ using a graphing utility or other tools.

Step 4: Find inflection points

To find inflection points (i.e., where the second derivative changes sign), first compute the second derivative: $$f''(x) = 300x^4 - 720x^3 - 900x^2 + 1800x + 240$$ Now look for the points where $$f''(x)=0$$ using a graphing utility or other tools.

Step 5: Piece it all together in a graph

Now that we have found the key features of the function, we can sketch a complete graph. At this point, it's a good idea to use a graphing utility to plot the function and verify our findings. Make sure the graph displays the x-intercepts from Step 1, y-intercept from Step 2, critical points from Step 3, and inflection points from Step 4. Adjust the sketch accordingly based on the graphing utility's output.

Key Concepts

Critical Points

Critical points are essential for understanding the shape of a graph. They occur where the derivative of a function, denoted as \( f'(x) \), equals zero or is undefined. In simpler terms, critical points are where the slope of the tangent to the curve is zero, resulting in horizontal tangents, or where the function is not differentiable.

To find the critical points for the polynomial \( f(x) = 10x^6 - 36x^5 - 75x^4 + 300x^3 + 120x^2 - 720x \), we first find its derivative:

• The derivative is \( f'(x) = 60x^5 - 180x^4 - 300x^3 + 900x^2 + 240x - 720 \).
• Next, set \( f'(x) = 0 \) and solve for \( x \) using algebraic methods or tools like graphing calculators.

Finding these points helps identify local maxima and minima, which are turning points on the graph. These points are significant as they help sketch the overall shape of the polynomial graph.

Inflection Points

Inflection points are where the function changes its curvature, moving from concave to convex or vice versa. This happens when the second derivative, \( f''(x) \), changes sign. Analyzing inflection points provides valuable information about the graph's behavior and its concavity.

To find the inflection points for our function \( f(x) \), you start by computing the second derivative:

• The second derivative is \( f''(x) = 300x^4 - 720x^3 - 900x^2 + 1800x + 240 \).
• Set \( f''(x) = 0 \) and solve for \( x \) using graphing utilities or solve algebraically.

Identifying inflection points assists in understanding how the graph bends and can help in sketching a more accurate graph of the polynomial.

X-intercepts

X-intercepts are crucial in graphing as they denote where the graph crosses the x-axis. These points are found by setting the function equal to zero and solving for \( x \). Essentially, at the x-intercepts, the value of \( y \) is zero.

For the polynomial \( f(x) = 10x^6 - 36x^5 - 75x^4 + 300x^3 + 120x^2 - 720x \):

• Factor the polynomial by extracting the common factor: \( f(x) = x(10x^5 - 36x^4 - 75x^3 + 300x^2 + 120x - 720) \).
• Use tools like graphing calculators to find the roots of the remaining polynomial to get all possible x-intercepts.

These intercepts are important as they are the points where the curve touches or crosses the x-axis, providing a solid foundation for sketching a graph of the function.

Y-intercepts

Y-intercepts represent the points where the graph crosses the y-axis. They are found by substituting \( x = 0 \) into the function, which simplifies to finding the constant term in the polynomial. At these points, \( x \) is zero, hence they mark the starting or ending point on the y-axis.

For the polynomial \( f(x) = 10x^6 - 36x^5 - 75x^4 + 300x^3 + 120x^2 - 720x \):

• Substitute \( x = 0 \) into the function: \( f(0) = 0 \).
• This calculation shows that the y-intercept is the point (0,0).

The y-intercept is vital as it gives a reference point on the graph and helps when sketching the overall curve of the function.