Target 6.1 The student will be able to use l Hôpital s Rule to evaluate indeterminate limits. lim. lim. 0, then

Transcription

1 Target 6.1 The student will be able to use l Hôpital s Rule to evaluate indeterminate limits. Recall from Section 2.1 Indeterminate form is when lim. xa g( Previously, we tried to reduce and then re-evaluate the limit to get an answer. However, not all of these problems can be algebraically reduced. L Hopital s Rule If f ( g(, f ( and g ( exist, and g (, then exists. lim g( lim f ( g( f ( g( if the second limit Note: This is NOT the quotient rule!!!! If evaluating the limit of the derivative yields another indeterminate answer, repeat the derivative process until you get a numerical answer. L Hopital s Rule also applies to lim xa g ( x ) that result in other indeterminate forms such as,, and. L Hopital s Rule with Natural Logs If a limit results in 1,, or, use ln L xa lim so that ln ) lim e f ( x L lim e. Target 6.2 The student will be able to use differentials to approximate the change in function values. Differentials The differential dy f ( dx is a dependent variable. It indicates the amount of change in the y- coordinate in terms of the change in the x-coordinate.

2 Target 6.3 The student will be able to solve physics problems involving motion along a line, position, velocity, and acceleration using algebraic and graphical representations. Physics Applications of Calculus Slope of a position vs. time graph = velocity (for the graph shown, slope = velocity in m s ) Therefor the derivative of position is velocity d (position) = velocity dx Slope of a velocity vs. time graph = acceleration (for the graph shown, slope = acceleration in Therefor the derivative of velocity is acceleration 2 m s ) d (velocity) = acceleration dx In Calculus terms: s( position s( v( velocity s( v( a( accelerati on Speed = Velocity The speed of an object is increasing if both v ( and a ( have the same sign (i.e., both are positive or both are negative). The speed of an object is decreasing if v ( and a ( have opposite signs (i.e., one is positive and the other is negative).

3 Target 6.4 The student will be able to solve optimization problems. Optimization refers to situations in which you want to maximize profit while minimizing cost, maximize an enclosed area while minimizing the amount of fence, maximize the volume of a container while minimizing the amount of material needed for the container (surface are, etc. Strategy for Solving Optimization Problems 1. Understand the problem. Read the problem carefully. Identify the information you need to solve the problem. 2. Develop a mathematical model of the problem. Draw pictures and label the parts that are important to the problem. Use variables to represent the quantities to be maximized or minimized. Using those variables, write a function whose extreme values (min/ma will give the information needed. 3. Find the domain of the function. (Graphing the function may help.) Determine what values of the variable(s) make sense in the problem. 4. Identify critical points and endpoints. Find where the derivative is zero or fails to exist. 5. Solve your mathematical model. If unsure of the result, support or confirm your solution with another model. 6. Interpret the solution. Translate your mathematical result into the context of the problem and decide whether the result makes sense.

4 Target 6.5 The student will be able to solve problems involving related rates. Strategy for Solving Related Rates Problems 1. Understand the problem. Read the problem carefully. Identify the variable whose rate of change you seek and the variable(s) whose rate(s) of change you know. 2. Develop a mathematical model of the problem. Draw a picture and label the parts that are important to the problem. Be sure to distinguish constant quantities from variables that change over time. Only constant quantities can be assigned numerical values at the start. 3. Write an equation relating the variable whose rate of change you seek with the variable(s) whose rate(s) of change you know. The formula is often geometric, but it could come from a scientific application. 4. Differentiate both sides of the equation with respect to time t. Be sure to follow all differentiation rules. The chain rule is especially critical. 5. Substitute values for any quantities that depend on time. Notice that it is only safe to do this after the differentiation step. Substituting too soon will result in an incorrect answer. 6. Interpret the solution. Translate your mathematical result into the context of the problem and decide whether the result makes sense. Appropriate units are necessary on all answers. * Recall of geometry formulas and relationships will be critical in this section!

5 Unit Six References Essential Skill Target 6.1 The student will be able to use l Hôpital s Rule to evaluate indeterminate limits. Text Examples p.45 #1-1, Target 6.2 The student will be able to use differentials to approximate the change in function values. p.242 #19-26 Target 6.3 The student will be able to solve physics problems involving motion along a line, position, velocity, and acceleration using algebraic and graphical representations. p.135 #1-6, 8-14, 16, 18-24, 27-28, 37 Target 6.4 The student will be able to solve optimization problems. p.226 #1-13, 18-21, 23-27, 3, 33, 35 Target 6.5 The student will be able to solve problems involving related rates. p.251 #1-29, 31-35