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find the inverse of Β f(x) = 5/2 βˆ’ x, x < 2 ;

1/x, x β‰₯ 2


a) Find the indefinite integral of the function

𝑦 = 3𝑑2 + 2𝑒3𝑑 + 2 + 2 cos 3𝑑


6. Do the following tasks using Mathematica.

Given,

f(x) = x 3 βˆ’ x 2 βˆ’ 2x + 1

g(x) = sin(x)

(a) Plot the above functions in a single graph for βˆ’Ο€ ≀ x ≀ Ο€.

(b) Find the limits of the integration for the the area of the region enclosed by f(x) and g(x) for βˆ’Ο€ ≀ x ≀ Ο€.

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+7xβˆ’4) dx

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

5. Evaluate in terms of beta function: "\\int_{0}^{1}" x2 (1 βˆ’ 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 Ο€.

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 Ο€, 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

{F} Calculate the approximate value of 10 to four decimal places by taking the first four


terms of an appropriate Taylor’s series.


{F} The equation for a displacement 𝑠(π‘š), at a time 𝑑(𝑠) by an object starting at a displacement of 𝑠0 (π‘š), with an initial velocity 𝑒(π‘šπ‘  βˆ’1 ) and uniform acceleration π‘Ž(π‘šπ‘  βˆ’2 ) is: 𝑠 = 𝑠0 + 𝑒𝑑 + 1 2 π‘Žπ‘‘ 2 A projectile is launched from a cliff with 𝑠0 = 30 π‘š, 𝑒 = 55 π‘šπ‘  βˆ’1 and π‘Ž = βˆ’10 π‘šπ‘  βˆ’2 . The tasks are to: a) Plot a graph of distance (𝑠) vs time (𝑑) for the first 10s of motion. b) Determine the gradient of the graph at 𝑑 = 2𝑠 and 𝑑 = 6𝑠. c) Differentiate the equation to find the functions for: i) Velocity (𝑣 = 𝑑𝑠 𝑑𝑑) ii) Acceleration (π‘Ž = 𝑑𝑣 𝑑𝑑 = 𝑑 2 𝑠 𝑑𝑑2 ) d) Use your results from part c to calculate the velocity at 𝑑 = 2𝑠 and 𝑑 = 6𝑠. e) Compare your results for part b) and part d). f) Find the turning point of the equation for the displacement 𝑠 and using the second derivative verify whether it is a maximum, minimum or point of inflection. g) Compare your results from f) with the graph you produced in a).


Let F(x,y)=2x^2+3sin y and let G(x,y)=e^x-2y.Compute the Jacobian determinant d(F ,G)/d(x,y)

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