Mathematics Foundations/9.3 Applications of Derivatives

9.3 Applications of Derivatives Introduction Derivatives are powerful mathematical tools that have numerous practical applications beyond just calculating rates of change. This section explores how derivatives can be used to solve various real-world problems and analyze functions in depth. Related Rates Related rates problems involve finding how one quantity changes with respect to another when both quantities are related through an equation and changing with time. General Approach 1. Identify all variables and what is given 2. Write an equation relating the variables 3. Differentiate both sides of the equation with respect to time 4. Substitute known values and solve for the unknown rate Example: Expanding Circle A circular oil slick is expanding at a rate of 2 m²/min. How fast is the radius increasing when the radius is 3 meters? Solution: The area of a circle is Differentiating with respect to time: Given information: m²/min m Solving for : m/min Mean Value Theorem The Mean Value Theorem states that if a function is continuous on the closed interval and differentiable on the open interval , then there exists at least one point in such that: Geometric Interpretation The theorem guarantees that there is at least one point where the tangent to the curve is parallel to the secant line connecting the endpoints and . Example For on , find the value of that satisfies the Mean Value Theorem. Solution: Setting : Monotonicity and Extrema Derivatives help determine where functions are increasing, decreasing, and have local extrema. First Derivative Test If on an interval, then is increasing on that interval If on an interval, then is decreasing on that interval If and changes sign at , then is a local extremum Second Derivative Test For a critical point where : If , then is a local minimum If , then is a local maximum If , the test is inconclusive Example: Analyzing a Function Find the intervals of increase/decrease and local extrema for . Solution: Critical points: and when or (function increasing) when (function decreasing) , so is a local maximum , so is a local minimum Curve Sketching Derivatives provide essential information for sketching accurate graphs of functions. Process 1. Find the domain of the function 2. Find any symmetry (even, odd, periodic) 3. Find intercepts (x and y) 4. Find asymptotes (horizontal, vertical, slant) 5. Find critical points and intervals of increase/decrease 6. Find inflection points and intervals of concavity 7. Sketch the graph using all information Example Sketch the graph of . Solution: Domain: All real numbers except where denominator equals zero, which never happens. Symmetry: , so the function is even. Intercepts: , so (0,0) is the only intercept. Asymptotes: As , , so is a horizontal asymptote. Critical points: when when (increasing) when (decreasing) Simplifying: Inflection points: when L'Hôpital's Rule L'Hôpital's Rule is used to evaluate limits that result in indeterminate forms. The Rule If results in the indeterminate form or , then: provided the limit on the right exists or is infinite. Example Evaluate . Solution: This limit has the form when . Applying L'Hôpital's Rule: Linear Approximation Linear approximation uses the tangent line at a point to approximate function values nearby. Formula The linear approximation of near is: Example Find the linear approximation of near , and use it to approximate . Solution: Linear approximation: Approximating : Newton's Method Newton's Method is an iterative technique for finding roots of equations. Algorithm Starting with an initial guess , generate the sequence: This sequence often converges to a root of . Example Use Newton's Method to approximate by finding a root of . Solution: Newton's iteration: Starting with : The actual value is , so we have a good approximation. Exercises 1. A spherical balloon is being inflated at a rate of 3 cubic inches per second. How fast is the radius increasing when the radius is 5 inches? 2. Prove that the function has exactly one real root. 3. Find the absolute maximum and minimum values of on the interval . 4. Use linear approximation to estimate . 5. Find the equation of the tangent line to the curve at the point where . 6. Use Newton's Method to approximate the positive root of to three decimal places. 7. Evaluate using L'Hôpital's Rule. 8. A particle moves along a straight line with position function where is measured in meters and in seconds. Find all times when the particle is at rest. 9. Sketch the graph of , showing all important features. 10. A rectangular box with no top is to be constructed from 12 square feet of cardboard. Find the dimensions that maximize the volume.