Calculus Answers

(a) let f : R → R be a function defined by
f(x) = x + 4 if x ≤ 1
ax + b if 1 < x ≤ 3
3x − 8 if x > 3
Find the values of a and b that makes f(x) continuous on R.

.(a) Consider a function h(u,v)=4u^2+4v^2+1. Find the level curves for k=n+1,n+2,n+3,n+4,n+5,n+6,n+7.Where n is your arid number (for example if your arid number is 19-arid-12345 then choose n=12345). Perform all steps clearly in detail. Also Sketch the neat Graph. (3 points)
(b) Let u=f(v),where f(v)=b(v^2-2v) draw the graph for different values of b∈[-5,5]. Just imagine that the values of b lies on the axis which is perpendicular to the laptop screen. Explain the behavior of graph when we change the values of b by assuming that b denotes the graph distance in back or in front of laptop screen? Also explain about the shape of graph (how it look like)? What are the necessary conditions for different types of functions to be defined in two variables explain with examples? How the conditions affect the input and output variables?

Find the volume of the solid generated by revolving the region bounded by the curves y = x^2 and y = 4x − x^2 about the line y = 6.

Sketch the graph of a continuous function f(x) satisfying the following properties:
(i) the graph of f goes through the origin
(ii) f'(−2) = 0 and f'(3) = 0.
(iii) f'(x) > 0 on the intervals (−∞, −2) and (−2, 3).
(iv) f'(x) < 0 on the interval (3, ∞).
Label all important points.

Suppose a particle P is moving in the plane so that its coordinates are given by P(x, y),
where x = 4 cos 2t, y = 7 sin 2t.
(i) By finding a, b ∈ R such that x^2/a^2+y^2/b^2= 1, show that P is travelling on an elliptical path.
(ii) Let L(t) be the distance from P to the origin. Obtain an expression for L(t).
(iii) How fast is the distance between P and the origin changing when t = π/8?

Find the volume of the solid generated by revolving the region bounded by the curves y =x2 and y =4x−x2 about the line y = 6.

Verify that the given family of functions solves the differential equation. (i)
dy dt
(ii)
dy dt
= (1 − 2t)y2, y = (c) Evaluate the integral
= y2 sin t, y = 1
0 ( 1
C − t + t2 . 1
C + cos t x2
√ . 4 − x2)3 dx

Differentiate the following functions with respect to x:
(i) ln(1 + sin^(2) x)
(ii) x^x
.

Evaluate the following integrals:
(i) ∫ x ln(x + 1)dx

(ii) ∫ ((sin³ (In x) cos²(In x)) / (x)) dx