Calculus Answers

6. Do the following tasks using Mathematica.

Given,

f(x) = x 3 − x 2 − 2x + 1

g(x) = sin(x)

(a) Plot the above functions in a single graph for −π ≤ x ≤ π.

(b) Find the limits of the integration for the the area of the region enclosed by f(x) and g(x) for −π ≤ x ≤ π.

Hint: Solve the equations to find the intersections.

(c) Finally, do the integration to find the area.

1. Find the area under the curve y = 3x+ 1 over the interval [2, 6] using right end point approximation.

2. Evaluate the integral and state whether it converges or diverges: "\\int_{-\u221e}^{0}" e 3x dx

3. Integrate the rational function by partial fraction (decomposition): ∫ (11x+17) / (2x2+7x−4) dx

4. Evaluate the integral by trigonometric substitution: ∫ dx / (1−x²)^3/2

5. Evaluate in terms of beta function: "\\int_{0}^{1}" x² (1 − x³ )^3/2 dx

{F} 1. Consider the graph of the function y = sin x + cos x. Describe its overall shape.

• Is it periodic?
• How do you know?

2. Using a graphing calculator or other graphing device, estimate the x- and y-values of the maximum point for the graph (the first such point where x > 0). It may be helpful to express the x-value as a multiple of π.

3. Now consider other graphs of the form y = A sin x + B cos x for various values of A and B.

• Sketch the graph when A = 2 and B = 1, and, find the x - and y-values for the maximum point. (Remember to express the x-value as a multiple of π, if possible.)
• Has it moved?

4. Repeat and sketch the graph for A = 1, B = 2.

• Is there any relationship to what you found in part (2)?

5. Explain what you have discovered from completing this activity using details and examples.

{F} Trace the following curves:-

(ii) x = a cos3(t) ; y = b sin3(t)

(iii) x =3at/1+t3; y =3at2/1+t3

find the domain and range of  H (x) = sqrt(x2 − 4/ x − 2)