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find the inverse of Β f(x) = 5/2 β x, x < 2 ;
1/x, x β₯ 2
a) Find the indefinite integral of the function
π¦ = 3π‘2 + 2π3π‘ + 2 + 2 cos 3π‘
6. Do the following tasks using Mathematica.
Given,
f(x) = x 3 β x 2 β 2x + 1
g(x) = sin(x)
(a) Plot the above functions in a single graph for βΟ β€ x β€ Ο.
(b) Find the limits of the integration for the the area of the region enclosed by f(x) and g(x) for βΟ β€ x β€ 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): β« (11x+17) / (2x2+7xβ4) dx
4. Evaluate the integral by trigonometric substitution: β« dx / (1βx2)3/2
5. Evaluate in terms of beta function: "\\int_{0}^{1}" x2 (1 β 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 Taylorβs series.
{F} The equation for a displacement π (π), at a time π‘(π ) by an object starting at a displacement of π 0 (π), with an initial velocity π’(ππ β1 ) and uniform acceleration π(ππ β2 ) is: π = π 0 + π’π‘ + 1 2 ππ‘ 2 A projectile is launched from a cliff with π 0 = 30 π, π’ = 55 ππ β1 and π = β10 ππ β2 . The tasks are to: a) Plot a graph of distance (π ) vs time (π‘) for the first 10s of motion. b) Determine the gradient of the graph at π‘ = 2π and π‘ = 6π . c) Differentiate the equation to find the functions for: i) Velocity (π£ = ππ ππ‘) ii) Acceleration (π = ππ£ ππ‘ = π 2 π ππ‘2 ) d) Use your results from part c to calculate the velocity at π‘ = 2π and π‘ = 6π . e) Compare your results for part b) and part d). f) Find the turning point of the equation for the displacement π 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 β 4/ x β 2)
