Related Rates Examples - Calculus I - Lecture Notes | MA 181, Study notes of Calculus

Download Related Rates Examples - Calculus I - Lecture Notes | MA 181 and more Study notes Calculus in PDF only on Docsity! h 13 ft x MA181 4.1 Related Rates The idea: If we can find a formula relating two quantities then by taking the derivative of both sides (with respect to some third quantity, usually time) we have a formula relating their derivatives (rates). Example 1 Air is being pumped in to a balloon so that its volume is increasing at a rate of 100 cm3/s. How fast is the radius of the balloon increasing when the diameter is 50 cm? Example 2 A 13 ft ladder is resting against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 5 ft from the wall? h 4 2 Example A water tank has the shape of an inverted circular cone with base radius 2 m and height 4 m. If water is being pumped in to the tank at the rate of 2 m3/min, find the rate at which the water level is rising when the water is 3 m deep. Example 4 Car A is travelling west at 50 mi/hr and car B is travelling north at 60 mi/hr. Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 0.3 mi and car B is 0.4 mi from the intersection? How do we find local extrema? Fermat's Theorem: Definition of a critical number: Example 5 Find the critical numbers for ( ) ( )2/3 2f x x x= + Fermat's Theorem Revised Strategy for finding absolute extrema on a closed interval Example 6 Find the exact maximum and minimum values of ( ) 2sinf x x x= − over the interval 0 2x π≤ ≤ and check them on your graphing calculator. HW 4.2 # 3, 5, 7, 11, 13, 15, 17, 21, 23, 29, 31, 37, 39, 43, 45, 53 MA181 4.3 Derivatives and the Shapes of Curves Mean Value Theorem: Application to velocity: Increasing/Decreasing Test Example 1 Find where the function ( ) 4 3 23 4 12 5f x x x x= − − + is increasing or decreasing. The First Derivative Test Find the local maximum and minimum values of the function from the previous example. Concavity Concavity Test Second Derivative Test Example 2 Use the second derivative test on the previous example MA181 4.4 Graphing with Calculus and Calculators Example 1 Graph the polynomial ( ) 6 5 3 22 3 3 2f x x x x x= + + − . Use the graphs of ( )f x′ and ( )f x′′ to estimate all maximum and minimum points and intervals of concavity. Example 2 Draw the graph of the function ( ) 2 2 7 3x xf x x + + = Find the maximum and minimum values, and the intervals of concavity. Example 3 Graph the function ( ) ( )( )sin sin 2f x x x= + and estimate all extrema and intervals of concavity. Example 4 Graph the curve with parametric equations ( ) 2 1x t t t= + + ( ) 4 3 23 8 18 25y t t t t= − − + in a viewing rectangle that displays all the important features of the curve and find the coordinates of all interesting points on the curve. HW 4.4 # 1, 7, 9, 11, 21 MA181 4.6 Optimization Problems Example 1 Find the dimensions of a rectangle with perimeter 100 ft whose area is as large as possible. Example 2 A farmer has 2400 ft of fence and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? Example 3 A cylindrical can is to hold 1 L of oil. Find the dimensions of the can that will minimize the cost of the metal to manufacture the can.. MA181 4.8 Newton's Method How do we solve equations like: 5 3 24 3 7 2 0x x x x+ − + − = or cos x x= The idea 1) Turn it in to an equation set equal to zero so that we are finding roots 2) Make an initial guess 0x 3) Find the tangent line at this point 4) Find the value of x for which the tangent line is 0 5) Use this solution as the next guess to go back to step 3 6) Stop when you see agreement to as many decimal places as you need Let's try to get a formula for this process: Example 1 Use 0 2x = and find 3x for the equation 3 2 5 0x x− − = Example 2 Use Newton's Method to estimate 6 2 to 8 decimal places. Example 3 Estimate the solution to cos x x= to 6 decimal places. 4.8 # 5, 7, 9, 11, 13 MA181 4.9 Antiderivatives Definition of an antiderivative: Theorem: Example 1 Find the general antiderivative of each of the following a) cos x b) 2x c) , 1nx n ≠ − d) 1 x Antidifferentiation Table We let F f′ = and G g′ = Function Particular antiderivative Function Particular Antiderivative ( ) c f x sin x ( ) ( )f x g x+ 2sec x , 1nx n ≠ − sec tanx x 1 x 2 1 1 x− xe cos x 2 1 1 x+