Calculus Question

Question

Consider the function .

(A) On your own graph paper, using an appropriate scale for your axes, sketch the graph of f(x) and shade the region bounded by , the line x = -1, the line x = 4, and the x-axis. In the blank below, name the shape of the graph produced.  

(B) Approximate the area of the region you shaded in part (A) using a left-endpoint Riemann sum with five equal intervals. 

 

(C) Write and evaluate a definite integral that represents the area you shaded in part (A). Enter the value in the blank below. (You will not receive credit if you do not show the integral expression in your submitted work.)

 

(D) Find the average slope of f(x) on [-4, 1].

 

(E) Find the value or values of c guaranteed by the Mean Value Theorem for Integrals for  on the interval [-1, 4]. (If there is more than one, list them in order from least to greatest, separated by commas.)

 

Round any decimal answers to the thousandths place, if necessary.

Answer 1

You should not expect a tutor to answer all the parts of this question; you need to work on the problem yourself!

I will start you off.

The graph is a parabola opening downward.

The area involved is not all above the x axis...if you do a straight integration, you will get an algebraic area....and, in fact, I would ask what is really wanted before attempting to answer this part of the question or part B.

Part D is just [f(4)-f(-1)]/5

For the last part you need to find where on the interval the derivative of f(x) equals the answer in part D.