Math Problem Statement
To graph the curve
y=4+5x2−2x3y = 4 + 5x^2 - 2x^3 y=4+5x2−2x3
and the tangent lines
y=4x+3y = 4x + 3 y=4x+3
and
y=−4x+16y = -4x + 16 y=−4x+16
, plot the curve and the lines on a common axis. The curve will be a cubic function, and the lines will intersect the curve at the points of tangency found above.
Solution
Let's graph the cubic function along with the given tangent lines on a common axis.
Given functions:
- Cubic function:
- Tangent lines:
I'll generate the plot now. numpy as np import matplotlib.pyplot as plt
Define the cubic function
def cubic_function(x): return 4 + 5x**2 - 2x**3
Define the tangent lines
def tangent1(x): return 4*x + 3
def tangent2(x): return -4*x + 16
Generate x values
x_values = np.linspace(-2, 4, 400) # Covering a reasonable range
Compute y values for the cubic function and tangent lines
y_cubic = cubic_function(x_values) y_tangent1 = tangent1(x_values) y_tangent2 = tangent2(x_values)
Plot the cubic function
plt.figure(figsize=(8, 6)) plt.plot(x_values, y_cubic, label=r'$y = 4 + 5x^2 - 2x^3$', color='blue', linewidth=2)
Plot the tangent lines
plt.plot(x_values, y_tangent1, label=r'$y = 4x + 3$', color='red', linestyle='dashed', linewidth=2) plt.plot(x_values, y_tangent2, label=r'$y = -4x + 16$', color='green', linestyle='dashed', linewidth=2)
Highlight the points of tangency (solved analytically or via computation earlier)
tangent_points_x = [1, 2] # Example points of tangency (you might want to verify these) tangent_points_y = [cubic_function(x) for x in tangent_points_x]
plt.scatter(tangent_points_x, tangent_points_y, color='black', zorder=3, label='Points of Tangency')
Labels and legend
plt.xlabel("x") plt.ylabel("y") plt.axhline(0, color='black', linewidth=0.5) plt.axvline(0, color='black', linewidth=0.5) plt.legend() plt.title("Graph of the Cubic Function and Tangent Lines") plt.grid(True)
Show the plot
plt.show()Here is the graph of the cubic function along with the given tangent lines. The black dots indicate the points of tangency. Let me know if you want further analysis or a breakdown of how these points were found.
Related Questions:
- How do we find the points of tangency algebraically?
- What is the derivative of the cubic function, and how does it relate to the tangent lines?
- How can we verify that these lines are indeed tangents?
- What happens to the shape of the cubic function if we change the coefficients?
- Can we determine additional tangent lines at different points?
Tip:
To find points of tangency, set the derivative of the function equal to the slope of the given line and solve for .
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Math Problem Analysis
Mathematical Concepts
Cubic Functions
Tangents
Derivatives
Graphing
Formulas
y = 4 + 5x^2 - 2x^3
y = 4x + 3
y = -4x + 16
Theorems
Derivative of a cubic function
Tangent line equation
Suitable Grade Level
Grades 11-12
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