First Principles Derivation of Velocity and Acceleration
A particle’s position along a straight line is given by \(s(t) = 4t^3 – 2t^2 + t\), where \(s\) is in meters and \(t\) in seconds. Answer the following:
Part b
Find the second derivative \(f”(x)\) and evaluate it at each critical point found in part (a).
Part a
For the cubic function \(f(x)=ax^3+bx^2+cx+d\), solve \(f”(x)=0\) to find the x-coordinate of the potential inflection point.
Concavity and Inflection Point Analysis
Given the function \(f(x)=x^3-6x^2+9x+15\), determine the intervals of concavity and identify any points of inflection by analyzing the sign chart of the second derivative, \(f”(x)\).
Determining Inflection Points of a Function
Consider a function \(y = f(x)\) which is twice differentiable. Inflection points occur where the concavity of the function changes, indicated by the second derivative \(\frac{d^2y}{dx^2}\). (a) Solve \(\frac{d^2y}{dx^2} = 0\) to determine the candidate value(s) for an inflection point. (b) Verify that a sign change occurs in \(\frac{d^2y}{dx^2}\) around the candidate value(s) to confirm […]
Analyzing Concavity via Second Derivative
Complete the following two parts:\n\n(a) SL: For the function \(f(x)=2x^3 – 3x^2 + x – 5\), compute its second derivative \(f”(x)\).\n\n(b) HL: For the function \(g(x)=\frac{2x^2-8}{x-2}\), determine the intervals where \(g(x)\) is concave upward and concave downward based on its second derivative, and state any inflection points. Be mindful of the domain of \(g(x)\).
Analysis of Polynomial Turning Points
Consider the polynomial function \(f(x)= x^4 – 4x^2 + 1\). Sketch its graph, clearly marking the turning points and points of inflection. Determine the coordinates of the turning points and state the intervals on which \(f(x)\) is increasing and decreasing.
Part b
Using your result from part (a), find the second derivative \(\frac{d^2y}{dx^2}\) in terms of \(t\).
Parametric Second Derivative of Trigonometric and Exponential Functions
Consider the parametric functions \(x(t)=\cos(t)\) and \(y(t)=e^t\). (a) Find the first derivative \(\frac{dy}{dx}\) in terms of \(t\). (b) Using your result from part (a), compute the second derivative \(\frac{d^2y}{dx^2}\) in terms of \(t\), ensuring your answer is fully expressed in terms of \(t\).