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#Answer

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]

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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.

2

3

(a) Find the Maclaurin series for $e^x$ and prove that it represents $e^x$ for all $x$ .

(b) Define Initial Value Problem. Solve that value problem of $y^2 + 5y = 1, y(0) = 2$

(c) Find the volume of a sphere of radius $r$

4

Section B

Answer any eight questions.

4

If f(x) = √x and g(x) = √(3 − x), find g ∘ f and g ∘ g.

5

Use Continuity to evaluate the limit

6

Verify Mean Value Theorem by f(x) = x3 – 3x + 3 for [-1, 2]

7

Sketch the curve y = x³+ x

8

Determine whether the integral is convergent or divergent:

9

Find the length of the arc of the semi cubical parabola y² = x²between the points (1,1) and (4,8)

10

Find the solution of y”+6y’+9=0, y(0)=2, y'(0)=1

11

Test the convergence of the series

12

13

Define limit of a function. Find limit:

14

Find the extremes values of f(x, y) = y² – x²

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.

2

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.

4

Section B

Answer any eight questions.

4 If f(x) = √x and g(x) = √(3 − x), find g ∘ f and g ∘ g.

5 Use Continuity to evaluate the limit

6 Verify Mean Value Theorem by f(x) = x3 – 3x + 3 for [-1, 2]

7 Sketch the curve y = x3 + x

8 Determine whether the integral is convergent or divergent:

9 Find the length of the arc of the semi cubical parabola y2 = x2 between the points (1,1) and (4,8)

10 Find the solution of y”+6y’+9=0, y(0)=2, y'(0)=1

11 Test the convergence of the series

12

13 Define limit of a function. Find limit:

14 Find the extremes values of f(x, y) = y2 – x2

1

a. A function is defined by

Colculate f(-1), f(3) and sketch graph.

b.

3

(a) Find the Maclaurin series for $e^x$ and prove that it represents $e^x$ for all $x$ .

(b) Define Initial Value Problem. Solve that value problem of $y^2 + 5y = 1, y(0) = 2$

(c) Find the volume of a sphere of radius $r$

4

If f(x) = √x and g(x) = √(3 − x), find g ∘ f and g ∘ g.

5

Use Continuity to evaluate the limit

6

Verify Mean Value Theorem by f(x) = x3 – 3x + 3 for [-1, 2]

7

Sketch the curve y = x³+ x

8

Determine whether the integral is convergent or divergent:

9

Find the length of the arc of the semi cubical parabola y² = x²between the points (1,1) and (4,8)

10

Find the solution of y”+6y’+9=0, y(0)=2, y'(0)=1

11

Test the convergence of the series

13

Define limit of a function. Find limit:

14

Find the extremes values of f(x, y) = y² – x²