Intersection Analysis of Quadratic and Exponential Functions
Consider the functions \(f(x)=x^2-3x+2\) and \(g(x)=e^x\). We aim to analyze the intersection points of their graphs. (a) Set up the equation that must be solved to find the x-coordinate(s) of the intersection point(s) of \(f(x)\) and \(g(x)\). (b) Solve the equation numerically to approximate one x-coordinate of an intersection point, rounding your answer to three […]
Part c
Sketch the graphs of the three functions on the same axes and discuss how the number and nature of intersection points reflect differences in their behavior. Mention any discrepancies between analytical and calculator-based approaches.
Mixed Derivative Analysis of Polynomial and Exponential Functions
This question involves two parts: (a) For the function \(f(x) = 3x^3 – 6x^2 + 2\), calculate the first derivative \(f'(x)\) and the second derivative \(f”(x)\). Then, determine all stationary points and classify each as a maximum, minimum, or point of inflection using the second derivative test. (b) For the function \(g(x) = 5e^x – […]
Part b
For the function \(g(x) = 5e^x – 10\), find the first derivative \(g'(x)\) and the second derivative \(g”(x)\). Based on these, discuss the behavior of \(g(x)\) in terms of growth and concavity. Justify your conclusions.
Understanding Limit Notation
Explain what is meant by the notation \(\lim_{x \to a} f(x) = L\). In your explanation, describe the relationship between \(f(x)\), \(a\), and \(L\) as implied by this limit statement.