Calculus_Optimization_Solutions
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Calculus Solutions for Optimization
Textbook Section 4.2: Optimization
1. Steps for finding the local/global maximum/minimum
To find the local/global maximum/minimum of a function \(y = f(x)\) on a closed domain \
(D\), follow these steps:
1. Find the first derivative of the function \(f'(x)\).
2. Find the critical points by setting the first derivative equal to zero and solving for \(x\).
3. Determine the concavity by evaluating the second derivative at the critical points.
4. Evaluate the function \(f(x)\) at the critical points and the endpoints of the domain \(D\).
5. Compare these values to find the absolute maximum and minimum on the domain.
2. Absolute maximum on the interval [0, 2]
The function \(f(x) = 2x^3 + 3x^2 - 12x + 4\) has critical points at \(x = -2\) and \(x = 1\).
The second derivative test indicates that there is a local maximum at \(x = -2\) (concavity is
downwards) and a local minimum at \(x = 1\) (concavity is upwards). By evaluating the
function at the endpoints and the critical point within the domain [0, 2], we find that the
function has an absolute maximum at \(x = 2\) with a value of 8.
3. Largest value of b and smallest value of c
Based on the calculations, the largest value of \(b\) on the interval [0, 2] is 8, which is the
maximum value of \(f(x)\) within this interval. The smallest value of \(c\) is -3, which
corresponds to the minimum value of \(f(x)\) at \(x = 1\).
Textbook Section 4.3: Optimization and Modelling
1. Circular pond optimization problem
To minimize the time to get from point A to point B, we need to set up an equation that
represents the total time. This time is the sum of the time swimming from point A to point C
and the time walking from point C to point B. We can swim at a speed of 2 km/h and walk at
a speed of 3 km/h. The equation will involve the distance from A to C (which can be found
using the arc length formula for a circle) and the distance from C to B (which is a straight
line).
2. Domain for the function from the previous question
The domain for the function from the previous question will depend on the geometry of the
pond and the positions of points A and B. Typically, it would include all possible points C on
the arc between A and B where the person can land while swimming from A and start
walking to B.
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