a) To find the equation of the tangent line to the function y = ln(x^2 - 4) at x = 3:
Calculate the derivative: y' = 2x / (x^2 - 4).
Evaluate y' at x = 3 to find the slope of the tangent line: m = 6/5.
Use the point-slope formula to derive the equation of the tangent line.
Answer: The equation of the tangent line is y - ln(3^2 - 4) = 6/5(x - 3).
b) To determine the equation of the tangent to the curve y = cos^2(2^x) at x = π/3:
Find the derivative: y' = -4^x * sin(2^x) * ln(2).
Evaluate y' at x = π/3.
Use the point-slope formula to obtain the equation of the tangent line.
Answer: The equation of the tangent line is y - cos^2(2^(π/3)) = -4^(π/3) * sin(2^(π/3)) * ln(2)(x - π/3).
c) For the function y = e^x, find the equation of the tangent line parallel to the line 2x - y = 5:
Identify the x-coordinate x = ln(2) where e^x has the same slope as the given line.
Use the point-slope formula to derive the equation of the tangent line.
Answer: The equation of the tangent line is y - e^(ln(2)) = 2(x - ln(2)).
d) To determine the intervals of increase and decrease for f(x) = x^3e^x:
Examine the sign of the derivative: f'(x) = x^2e^x(3 + x).
Analyze sign changes of the derivative in different intervals to find where the function is increasing or decreasing.
Answer: The function is increasing on the intervals (-3, 0) and (0, ∞) and decreasing on the interval (-∞, -3).
e) For the function f(x) = e^x - 2x, locate the critical point at x = ln(2):
Answer: The minimum value of f(x) occurs at x = ln(2).
f) To find the maximum and minimum point(s) for y = (1/2)sin(2x) = 3cos(x) + x on the interval 0 < x < π:
Find the derivative of y with respect to x: y' = cos (2x) - 3sin(x) + 1.
Set ' equal to zero to find the critical points: cos (2x) - 3sin(x) + 1 = 0.
Solve for a to find the critical values. You may need to use numerical methods or trigonometric identities to solve this equation.
Once you have the critical points, evaluate the original function u at these points to find the corresponding maximum and minimum values.
For f) If you provide the critical points or specify the values of x for which you want to find the maximum and minimum values, I can write a specific answer.