A(x) =SECTION 4.3: DERIVATIVES AND THE SHAPES OF CURVESSECTION 4.4: GRAPHING WITH CALCULUS AND CALCULATORSSECTION 4.8: Newton's MethodSECTION 4.9: ANTIDERIVATIVESSECTION 5.4: THE AREA FUNCTIONRecall that we can use the notation to denote the area under the curve between x = a and x = b.1. Let.a. Sketch the region bounded by the graph of , the x –axis, x = 1,and x = b, where b is any number greater than 1.b. Using geometry, compute.c. Similarly, compute.d. Using your answers to parts (b) and (c) as a guide, compute.e. We now define the area function as. What is A(2)? A(4)? A(1)? A(x)? [Hint: You found a formula for A(x) in part (d).]2. Let .a. Sketch the region bounded by the graph of, the x –axis, x = 0,and x = b, where b is any number greater than 0.b. Using geometry, compute.c. Similarly, compute.d. Using your answers to parts (b) and (c) as a guide, compute.e. We now define another area function as . What is B(2)? B(4)? B(0)? B(x)?f. We now define a third area function C(x) as . How does this change the region on your sketch?g. What is C(2)? C(4)? C(1)? C(x)?3. The Punchline:We have now computed three different area functions. Fill in the blanks:You should notice a very interesting fact about the derivatives of the area functions – a fundamentally beautiful property. What is it?4. We are going to define one final function, . The region looks like this:Don’t worry about trying to find a simple formula for. But, using our amazing fact, fill in our last blank:Not only is the fact that we’ve discovered a surprising one; as we shall see, it is also extremely useful.SECTION 5.4: EVALUATING DEFINITE INTEGRALSWe will now use Converge to graph these same accumulation functions for us.4.3 COMPUTER EXAMPLESSECTION 4.2: MAXIMUM AND MINIMUM VALUESExample 6, page 273: Use Derive to estimate the absolute minimum and maximum values of the function f x x x( ) sin  2 on the interval 0 2 x.Before starting, be sure Derive is in Exact Precision. To do this, choose Declare, Simplification Settings. Check the Mode under Precision and change it to Exact if necessary.Type the function x 2sinx on the Authoring Line and hit ENTER. Click on . If necessary, click on Window, Tile Vertically from the menu bar.Change the window: choose Set, Plot Range. The horizontal should go from 0 to 2pi with 8 intervals and the vertical should go from 2 to 8 with 5 intervals. Click on to graph the function. Switch to Trace mode by hitting the F3 function key. Click in the vicinity of the absolute minimum and then use the arrow keys to refine the position of the cross-hair. As you do, the coordinates of the cross-hair will be displayed on the bottom left of the graphing window.  What is the approximate absolute minimum of this function on this interval?Can you determine from the graph the x-value where this absolute minimum occurs?Click in the vicinity of the absolute maximum and then use the arrow keys to refine the position of the cross-hair. What is the approximate absolute maximum of the function on this interval?Can you determine from the graph the x-value where this absolute maximum occurs?In class, you should use Calculus to determine the values of the absolute extrema on this interval.664.3 GROUP EXERCISESSECTION 4.3: DERIVATIVES AND THE SHAPES OF CURVES Below is a list of questions that you should consider when graphing a function)(xfy . You already know how to determine the answer to some of the questions. With your group, go through the list of questions and place a check mark next to the questions that you now how to answer and write a strategy on how to answer those questions.1. What is the domain?2. What are the intercepts?3. Is there any symmetry?4. Are there any asymptotes?5. Where is the graph increasing? Decreasing?6. Are there any relative extrema?7. Where is the graph concave up? Concave down?8. Are there any inflection points?674.3 COMPUTER EXAMPLESSECTION 4.3: DERIVATIVES AND THE SHAPES OF CURVESExample: Analyze the graph of f xxx x( )( )342. Include in your analysis: domain, asymptotes, critical values, local extrema, inflection points and concavity. Give a careful and accurate sketch.We will use Derive to assist us in this analysis since it can differentiate functions and solve equations.1. First, author the function.Type the function (x+3)/(x(x^24)) on the Authoring Line and hit ENTER.2. Now plot the graph: Click on . If necessary, click on Window, Tile Vertically from the menu bar. Then click on again.3. Change the Range: Click on Set, Plot Range. The horizontal should go from 6 to 6 with 12 intervals and the vertical should go from –4 to 4 with 8 intervals.What is the domain of f x( )? What are the vertical asymptotes of f x( )? Is there a horizontal asymptote? If so, what is the equation?4. Obtain the first derivative:In the algebra window, click on then Simplify. Write f x'( ) here:5. Change the precision to Mixed: Declare, Simplification Settings, and change the Mode to Mixed. NOTE : The default mode in Derive is Exact and therefore Derive will only look for exact answers and will not return an answer if an exact value is not found. In Mixed mode, Derive will return either exact or approximate answers. Approximate mode will only give an answerif you give Derive an interval to look in. Thus, it is often advisable to change the Mode to Mixed at the start of the program.684.3 COMPUTER EXAMPLES6. Solve the equationf x'( ) 0; i. e. find the critical values. First click on f x( ). Click on , then Either and Real, then Solve. NOTE : Selecting Either instructs Derive to use algebraic methods to solve the equation and if those fail, it will use numeric solution methods and choosing Real, tells Derive that you only want Real (as opposed to complex or imaginary) solutions to the equation. Notice that Derive returns three meaningful answers. Write down all three critical values here:7. We now need to test the critical values. We could use the first derivative test but since we will look for inflection points and need the second derivative anyway, we will use the second derivative test. First, we need to find the second derivative.Click on f x( ). Then click on then Simplify. Write f x''( ) here:8. To use the Second Derivative Test on the critical values, we need to determine the sign of the second derivative at each critical value.Click on f x''( ). Then click on , type in