Math Problem Statement

Question content area top Part 1 The function ​f(x)equalsone half x e Superscript negative x is positive and negative on the interval ​[negative 1​,3​]. a. Sketch the function on the interval. b. Approximate the net area bounded by the graph of f and the​ x-axis on the interval using a​ left, right, and midpoint Riemann sum with nequals4. c. Use the sketch in part​ (a) to show which intervals of ​[negative 1​,3​] make positive and negative contributions to the net area. Question content area bottom Part 1 a. Choose the correct graph. A. -1 3 -2 1 x y

A coordinate system has a horizontal x-axis labeled from negative 1 to 3 in increments of 0.5 and a vertical y-axis labeled from negative 2 to 1 in increments of 0.25. A smooth curve starts at the point left parenthesis negative 1 comma negative 1.36 right parenthesis comma rises concave down passing through the origin to a maximum in Quadrant 1 at the point left parenthesis 1 comma 0.18 right parenthesis comma then falls concave up comma passing through the point left parenthesis 2 comma 0.14 right parenthesis before ending at the point left parenthesis 3 comma 0.07 right parenthesis . All coordinates are approximate. Your answer is correct.B. -1 3 -2 1 x y

A coordinate system has a horizontal x-axis labeled from negative 1 to 3 in increments of 0.5 and a vertical y-axis labeled from negative 2 to 1 in increments of 0.25. From left to right, a smooth curve starts at the point (negative 1, 0), rises concave down to a maximum in Quadrant 2, then falls concave down, passing through the points (1, negative 0.35) and (2, negative 0.82) before ending at the point (3, negative 1.39). All coordinates are approximate. C. -1 3 -1 2 x y

A coordinate system has a horizontal x-axis labeled from negative 1 to 3 in increments of 0.5 and a vertical y-axis labeled from negative 1 to 2 in increments of 0.25. From left to right, a smooth curve falls concave up from the point (negative 1, 1.36) and passes through the points (0, 0.5), (1, 0.18), and (2, 0.07) before ending at the point (3, 0.02). All coordinates are approximate. D. -1 3 -1 2 x y

A coordinate system has a horizontal x-axis labeled from negative 1 to 3 in increments of 0.5 and a vertical y-axis labeled from negative 1 to 2 in increments of 0.25. From left to right, a smooth curve falls concave up starting at left parenthesis negative 1 comma 1.36 right parenthesis to a minimum at the point left parenthesis 0 comma 0 right parenthesis comma then rises concave down comma passing through the point left parenthesis 1 comma 0.18 right parenthesis to a maximum at the point left parenthesis 2 comma 0.27 right parenthesis comma then falling concave up comma ending at the point left parenthesis 3 comma 0.22 right parenthesis . All coordinates are approximate. Part 2 b. The net​ area, approximated using the left Riemann​ sum, is    negative 1.04. ​(Type an integer or decimal rounded to two decimal places as​ needed.) Part 3 The net​ area, approximated using the right Riemann​ sum, is    0.39. ​(Type an integer or decimal rounded to two decimal places as​ needed.) Part 4 The net​ area, approximated using the midpoint Riemann​ sum, is    enter your response here.

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Math Problem Analysis

Mathematical Concepts

Calculus
Exponential Functions
Riemann Sums
Area under Curve

Formulas

f(x) = (1/2) x e^(-x)
Left Riemann Sum = Σ f(x_i) Δx using left endpoints
Right Riemann Sum = Σ f(x_i) Δx using right endpoints
Midpoint Riemann Sum = Σ f(midpoints) Δx

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Grades 11-12