Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

5. Graphical Applications of Derivatives

Applied Optimization

Calculus

5. Graphical Applications of Derivatives

Applied Optimization

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Multiple Choice

Find the minimum and maximum values of the function $f (x, y) = x^{2} + y^{2}$ subject to the constraint $x + y = 4$ .

Minimum:

8

, Maximum:

18

Minimum:

8

, Maximum: No maximum

Minimum:

8

, Maximum:

16

Minimum:

8

, Maximum:

32

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Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the minimum and maximum values of the function f(x, y) = x^2 + y^2, subject to the constraint x + y = 4. This involves optimization with constraints, which can be solved using methods like substitution or Lagrange multipliers.

Step 2: Rewrite the constraint equation x + y = 4 to express one variable in terms of the other. For example, solve for y: y = 4 - x. This substitution will allow us to reduce the problem to a single-variable function.

Step 3: Substitute y = 4 - x into the function f(x, y) = x^2 + y^2. This gives a new function in terms of x only: f(x) = x^2 + (4 - x)^2. Expand and simplify this expression to make it easier to analyze.

Step 4: Find the critical points of f(x) by taking the derivative of f(x) with respect to x, setting it equal to zero, and solving for x. This will help identify potential minimum and maximum values.

Step 5: Evaluate f(x) at the critical points and at the boundary points (if applicable) to determine the minimum and maximum values. Since the constraint x + y = 4 is linear, check the values of f(x, y) at specific points satisfying the constraint, such as (x, y) = (0, 4) and (x, y) = (4, 0).