Mathematics I
Community
Mathematics I
2074 Boards
Section A
Answer any two questions.
1
a. A function is defined by
Colculate f(-1), f(3) and sketch graph.
b.
10
Connect with us on Discord to become a contributor.
2
10
Connect with us on Discord to become a contributor.
3
(a) Find the Maclaurin series for ex and prove that it represents ex for all x.
(b) Define Initial Value Problem. Solve that value problem of y2 + 5y = 1, y(0) = 2.
(c) Find the volume of a sphere of radius r.
10
Connect with us on Discord to become a contributor.
4
10
Connect with us on Discord to become a contributor.
Section B
Answer any eight questions.
4
If f(x) = √x and g(x) = √(3 − x), find g ∘ f and g ∘ g.
5
1. Finding g∘f(x)
Given f(x) = √x and g(x) = √(3−x)
g(f(x)) = g(√x) = √(3 − √x)
∴ g∘f(x) = √(3 − √x)
Domain of g∘f(x):
- For f(x) = √x to be defined: x ≥ 0
- For g∘f(x) = √(3 − √x) to be defined: 3 − √x ≥ 0 → √x ≤ 3 → x ≤ 9
- Combining both conditions: 0 ≤ x ≤ 9
∴ Domain of g∘f(x) = [0, 9]
2. Finding g∘g(x)
Given g(x) = √(3−x)
g(g(x)) = g(√(3−x)) = √(3 − √(3−x))
∴ g∘g(x) = √(3 − √(3−x))
Domain of g∘g(x):
- For inner g(x) = √(3−x) to be defined: 3 − x ≥ 0 → x ≤ 3
- For g∘g(x) = √(3 − √(3−x)) to be defined: 3 − √(3−x) ≥ 0 → √(3−x) ≤ 3 → 3−x ≤ 9 → x ≥ −6
- Combining both conditions: −6 ≤ x ≤ 3
∴ Domain of g∘g(x) = [−6, 3]
5
Use Continuity to evaluate the limit
5
Connect with us on Discord to become a contributor.
6
Verify Mean Value Theorem by f(x) = x3 – 3x + 3 for [-1, 2]
5
Connect with us on Discord to become a contributor.
7
Sketch the curve y = x3 + x
5
Connect with us on Discord to become a contributor.
8
Determine whether the integral is convergent or divergent:
5
Connect with us on Discord to become a contributor.
9
Find the length of the arc of the semi cubical parabola y2 = x2 between the points (1,1) and (4,8)
5
Connect with us on Discord to become a contributor.
10
Find the solution of y”+6y’+9=0, y(0)=2, y'(0)=1
5
Connect with us on Discord to become a contributor.
11
Test the convergence of the series
5
Connect with us on Discord to become a contributor.
12
5
Connect with us on Discord to become a contributor.
13
Define limit of a function. Find limit:
5
Connect with us on Discord to become a contributor.
14
Find the extremes values of f(x, y) = y2 – x2
5
Connect with us on Discord to become a contributor.