Calculus Answers

(a) A bowl of water has a temperature of 50◦C. It is put into a refrigerator where the temper-
ature is 5◦C. After 0.5 hours, the water is stirred and its temperature is measured to be
20◦C. It is then left to cool further, take t in Newton’s law of cooling to be measured in
minutes. Use Newton’s law of cooling to predict when the temperature will be 10◦C.

(b) Verify that the given family of functions solves the differential equation.
(i) dy
dt = (1 − 2t)y
2
, y =
1
C − t + t
2
.
(ii) dy
dt = y
2
sin t, y =
1
C + cost
.
(c) Evaluate the integral Z 1
0
x
2
(
√
4 − x
2)
3
dx

(a) Find the solution of the initial-value problem and determine its interval of existence (i.e
domain of the resulting function y).
(i) dy
dt =
1+t
2
t
, y(t = 1) = 0.
(ii) (t + 1) dy
dt = 1 − y, y(t = 0) = 3. [Hint: Let A = −(±e
−C).]
(b) Evaluate the following integrals:
(i) Z
x ln(x + 1)dx
(ii) Z
sin3
(ln x) cos2
(ln x)
x
dx.

Evaluate the integeral
∫((2x³ -4x -8)/ (x⁴ -x³ +4x² -4x)) dx

(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.
(b) Find the derivative of f(x) = tan−1
1 − x
1 + x

.
(c) Find f
0
(x) using logarithmic differentiation, where f(x) = e
−3x
√
2x − 5
(6 − 5x)
4
.
(d) Evaluate the integral Z
(x
3 + 1)1/3x
5
dx.

(a) Verify that y = e
2x
sin x is a solution to the differential equation d
2y
dx2
− 4
dy
dx + 5y = 0.
(b) Differentiate the following functions with respect to x:
(i) ln(1 + sin2 x) (ii) x
x
.
(c) Evaluate the integral Z
2x
3 − 4x − 8
x
4 − x
3 + 4x
2 − 4x
dx.

Find f′(x) using logarithmic differentiation , where f(x) =( e⁻³ˣ √(2x-5)) / (6 -5x)⁴

Find the derrivative of
f(x) = tan⁻¹((1-x)/(1+x))

Differentiate the following functions with respect to x:
(i) ln(1 + sin2 x) (ii) x²

.(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.
(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? )

.(a) Use Numerical Method to find Limit of function h(u,v)=bu^α v^(1-α),where b=2.01,α=0.25. Find lim┬((u,v)→(n,n))⁡〖h(u,v).〗 Where n is your arid number (for example if your arid number is 19-arid-12345 then choose n=12345). Choose at least 7 most nearest values of n for both (u,v)(Don’t choose far values from n marks will be deducted in that case). Construct neat table and also perform all calculations. And check whether the limit exist or not? If yes then what’s the value of limit.
(b) Define f(0,0) in a way that extends f(u,v)=uv (u^2-v^2)/(u^2+v^2 ) to be continuous at origin? (Hint: firstly write piecewise define function). Discuss continuity in detail? Also define the condition when this function is discontinuous at (0,0)?