Maths Week 8 Graded Assignments IIT Madras

Maths week 8 graded assignment Complete Solutions Are Discussed In This Blog We Hope This Might Help You All In Matching Answers . Or For Some Others Reasons Not Able To Complete Graded Assignments

1.Match the given functions in Column A with the equations of their tangents at the origin $(0, 0)$ in column B and the plotted graphs and the tangents in Column C, given in Table M2W2G1.

ii)

\to

\to

\to

\to

iii)

\to

\to

iii)

\to

\to

\to

\to

No, the answer is incorrect.
Score: 0

Accepted Answers:

ii)

\to

\to

\to

\to

iii)

\to

\to

1 point

2.Consider the following two functions $f (x)$ and $g (x)$ .

$f (x) = ⎩ ⎨ ⎧ x ( x - 3 ) x ^{3} - 9 x if x \neq = 0, 3 3 if x = 0 0 if x = 3$

$g (x) = {∣ x ∣ ⌊ x ⌋ if x \leq 2 if x > 2$

Choose the set of correct options.

f (x)

is discontinuous at both

x = 0

and

x = 3

f (x)

is discontinuous only at

x = 0

f (x)

is discontinuous only at

x = 3

g (x)

is discontinuous at

x = 2

g (x)

is discontinuous at

x = 3

No, the answer is incorrect.
Score: 0

Accepted Answers:

f (x)

is discontinuous only at

x = 3

g (x)

is discontinuous at

x = 3

3.

1 point

Consider the graphs given below:

Choose the set of correct options from the below.

Curve 1 is both continuous and differentiable at the origin.

Curve 2 is continuous but not differentiable at the origin.

Curve 2 has derivative 0 at

x = 0

Curve 3 is continuous but not differentiable at the origin.

Curve 4 is not differentiable anywhere.

Curve 4 has derivative 0 at

x = 0

No, the answer is incorrect.
Score: 0

Accepted Answers:

Curve 1 is both continuous and differentiable at the origin.

Curve 2 has derivative 0 at

x = 0

Curve 3 is continuous but not differentiable at the origin.

1 point

4.Choose the set of correct options considering the function given below:

$f (x) = {x s in x 1 if x \neq = 0, if x = 0$

f (x)

is not continuous at

x = 0

f (x)

is continuous at

x = 0

f (x)

is not differentiable at

x = 0

f (x)

is differentiable at

x = 0

The derivative of

f (x)

x = 0

(if exists) is

0

The derivative of

f (x)

x = 0

(if exists) is

1

No, the answer is incorrect.
Score: 0

Accepted Answers:

f (x)

is continuous at

x = 0

f (x)

is differentiable at

x = 0

The derivative of

f (x)

x = 0

(if exists) is

0

1 point

5. Let $f$ be a polynomial of degree 5, which is given by

$f (x) = a_{5} x^{5} + a_{4} x^{4} + a_{3} x^{3} + a_{2} x^{2} + a_{1} x + a_{0} .$

Let $f^{'} (b)$ denote the derivative of $f$ at $x = b$ . Choose the set of correct options.

a_{1} = f^{'} (0)

5 a_{5} + 3 a_{3} = 2 1 (f^{'} (1) + f^{'} (- 1) - 2 f^{'} (0))

4 a_{4} + 2 a_{2} = 2 1 (f^{'} (1) - f^{'} (- 1))

None of the above.

No, the answer is incorrect.
Score: 0

Accepted Answers:

a_{1} = f^{'} (0)

5 a_{5} + 3 a_{3} = 2 1 (f^{'} (1) + f^{'} (- 1) - 2 f^{'} (0))

4 a_{4} + 2 a_{2} = 2 1 (f^{'} (1) - f^{'} (- 1))

6. Let $f$ be a differentiable function at $x = 3$ . The tangent line to the graph of the function $f$ at the point $(3, 0)$ , passes through the point $(5, 4)$ . What will be the value of $f^{'} (3)$ ?

No, the answer is incorrect.
Score: 0

Accepted Answers:

(Type: Numeric) 2

1 point

7. Let $f$ and $g$ be two functions which are differentiable at each $x \in R$ . Suppose that, $f (x) = g (x^{2} + 5 x)$ , and $f^{'} (0) = 10$ . Find the value of $g^{'} (0)$ .

No, the answer is incorrect.
Score: 0

Accepted Answers:

(Type: Numeric) 2

1 point

8. Consider a function $f : R \to R$ defined as $f (x) = {15 x ^{3} s i n 12 x + A s i n x B if x \neq = 0, if x = 0.$ If $f (x)$ is continuous at $x = 0$ , then find the value of $90 B - A$

No, the answer is incorrect.
Score: 0

Accepted Answers:

(Type: Numeric) -1704

1 point

9. The distance (in meters) traveled by a car after $t$ minutes is given by the function $d (t) = g (6 t^{3} + 5 t^{2} + 8 t + 5)$ , where $g$ is a differentiable function with domain $R$ . Find the instantaneous speed of the car after 5 min, where $g^{'} (920) = 2$

No, the answer is incorrect.
Score: 0

Accepted Answers:

(Type: Numeric) 1016

1 point

10. Consider the following two functions

$p : R \to R$

$p (t) = ⎩ ⎨ ⎧ t - 5 6 e ^{(t - 5)} - 6 6 6 (t^{2} - 25)^{l n ( t - 5 ) 1} if 0 \leq t < 5, t = 5 if t > 5$

and

$q : R \to R$

$q (t) = ∣ t (t - 5) (t - 8) ∣$ and the following statements (a function is said to be continuous (respectively differentiable) if it is continuous (respectively differentiable) at all the points in the domain of the function).

$Statement P:$ Both the functions $p (t)$ and $q (t)$ are continuous.
$Statement Q:$ Both the functions $p (t)$ and $q (t)$ are not differentiable.
$Statement R:$ $p (t)$ is continuous, $q (t)$ is differentiable.
$Statement S:$ $q (t)$ is continuous, $p (t)$ is not differentiable.
$Statement T:$ Neither $p (t)$ nor $q (t)$ is continuous.

Find the number of correct statements.

No, the answer is incorrect.
Score: 0

Accepted Answers:

(Type: Numeric) 2

1 point

11. Consider the following function

p : R \to R

p (t) = ⎩ ⎨ ⎧ t - 5 8 e ^{(t - 5)} - 8 8 8 (t^{2} - 25)^{l n ( t - 5 ) 1} if 0 \leq t < 5, t = 5 if t > 5

If linear function

L_{p} (t) = A t + B

denotes the best linear approximation of the function

p (t)

at the point

t = 1

, find the value of

e ^{- 4} - 1 - 8 (A + B)

No, the answer is incorrect.
Score: 0

Accepted Answers:

(Type: Numeric) 16

1 point

12. Consider the following function
$q : R \to R q (t) = ∣ t (t - 2) (t - 8) ∣.$

If $m$ is slope of the tangent of the function $q (t)$ at point $t = 2 3$ , find the value $m - 4 27$

No, the answer is incorrect.
Score: 0

Accepted Answers:

(Type: Numeric) -14

1 point

The population of a bacteria culture of type A in laboratory conditions is known to be a function of time of the form

$p : R \to R$
$p (t) = ⎩ ⎨ ⎧ t - 3 t ^{3} - 27 27 e ^{81} ( t - 3 ) 1 (e^{27 t} - e^{81}) if 0 \leq t < 3, t = 3 if t > 3$
where $p (t)$ represents the population (in lakhs) and $t$ represents the time (in minutes). The population of a bacteria culture of type B in laboratory conditions is known to be a function of time of the forM

$q : R \to R$
$q (t) = ⎩ ⎨ ⎧ (5 t - 9)^{t - 2 1} e^{4} t - 2 e ^{t + 2} - e ^{4} if 0 \leq t < 2, t = 2 if t > 2$
where $q (t)$ represents the population (in lakhs) and $t$ represents the time (in minutes).

Using the information above, answer questions 13,14 and 15.

Consider the following statements (a function is said to be continuous if it is continuous at all the points in the domain of the function).

Statement P:

Both the functions

p (t)

and

q (t)

are continuous.

Statement Q:

p (t)

is continuous, but

q (t)

is not.

Statement R:

q (t)

is continuous, but

p (t)

is not.

Statement S:

Neither

p (t)

nor

q (t)

is continuous.

Find the number of the correct statements.

No, the answer is incorrect.
Score: 0

Accepted Answers:

(Type: Numeric) 1

1 point

L_{p} (t) = A t + B

denotes the best linear approximation of the function

p (t)

at the point

t = 1

, then find the value of

2 A + B

No, the answer is incorrect.
Score: 0

Accepted Answers:

(Type: Numeric) 18

1 point

L_{p} (t) = e^{4} (A t + B) + C e^{5}

denotes the best linear approximation of the function

q (t)

at the point

t = 3

, then find the value of

A + B + C

No, the answer is incorrect.
Score: 0

Accepted Answers:

(Type: Numeric) -2

1 point

Maths Week 8 Graded Assignments IIT Madras

1.Match the given functions in Column A with the equations of their tangents at the origin (0,0)(0,0) in column B and the plotted graphs and the tangents in Column C, given in Table M2W2G1.

No, the answer is incorrect.Score: 0

Accepted Answers:

2.Consider the following two functions �(�)f(x) and �(�)g(x).

�(�)={�3−9��(�−3)if �≠0,33if �=00if �=3f(x)=⎩⎨⎧x(x−3)x3−9xif x=0,33if x=00if x=3�(�)={∣�∣if �≤2⌊�⌋if �>2g(x)={∣x∣⌊x⌋if x≤2if x>2Choose the set of correct options.

No, the answer is incorrect.Score: 0

Accepted Answers:

3.

No, the answer is incorrect.Score: 0

Accepted Answers:

4.Choose the set of correct options considering the function given below:

�(�)={��� ��if �≠0,1if �=0f(x)={xsin x1if x=0,if x=0

No, the answer is incorrect.Score: 0

Accepted Answers:

5. Let �f be a polynomial of degree 5, which is given by

�(�)=�5�5+�4�4+�3�3+�2�2+�1�+�0.f(x)=a5x5+a4x4+a3x3+a2x2+a1x+a0.Let �′(�)f′(b) denote the derivative of �f at �=�x=b. Choose the set of correct options.

No, the answer is incorrect.Score: 0

Accepted Answers:

6. Let �f be a differentiable function at �=3x=3. The tangent line to the graph of the function �f at the point (3,0)(3,0), passes through the point (5,4)(5,4). What will be the value of �′(3)f′(3)?

No, the answer is incorrect.Score: 0

Accepted Answers:

7. Let �f and �g be two functions which are differentiable at each �∈�x∈R. Suppose that, �(�)=�(�2+5�)f(x)=g(x2+5x), and �′(0)=10f′(0)=10. Find the value of �′(0)g′(0).

No, the answer is incorrect.Score: 0

Accepted Answers:

8. Consider a function �:�→�f:R→R defined as �(�)={sin⁡12�+�sin⁡�15�3if �≠0,�if �=0.f(x)={15x3sin12x+AsinxBif x=0,if x=0. If �(�)f(x) is continuous at �=0x=0 , then find the value of 90�−�90B−A

No, the answer is incorrect.Score: 0

Accepted Answers:

9. The distance (in meters) traveled by a car after �t minutes is given by the function �(�)=�(6�3+5�2+8�+5)d(t)=g(6t3+5t2+8t+5) , where �g is a differentiable function with domain �R . Find the instantaneous speed of the car after 5 min, where �′(920)=2g′(920)=2

No, the answer is incorrect.Score: 0

Accepted Answers:

10. Consider the following two functions

No, the answer is incorrect.Score: 0

Accepted Answers:

No, the answer is incorrect.Score: 0

Accepted Answers:

12. Consider the following function �:�→��(�)=∣�(�−2)(�−8)∣.q:R→Rq(t)=∣t(t−2)(t−8)∣.If �m is slope of the tangent of the function �(�)q(t) at point �=32t=23, find the value �−274m−427

No, the answer is incorrect.Score: 0

Accepted Answers:

No, the answer is incorrect.Score: 0

Accepted Answers:

No, the answer is incorrect.Score: 0

Accepted Answers:

No, the answer is incorrect.Score: 0

Accepted Answers:

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1.Match the given functions in Column A with the equations of their tangents at the origin $(0, 0)$ in column B and the plotted graphs and the tangents in Column C, given in Table M2W2G1.

No, the answer is incorrect.
Score: 0

2.Consider the following two functions $f (x)$ and $g (x)$ .

$f (x) = ⎩ ⎨ ⎧ x ( x - 3 ) x ^{3} - 9 x if x \neq = 0, 3 3 if x = 0 0 if x = 3$

$g (x) = {∣ x ∣ ⌊ x ⌋ if x \leq 2 if x > 2$

Choose the set of correct options.

No, the answer is incorrect.
Score: 0

No, the answer is incorrect.
Score: 0

$f (x) = {x s in x 1 if x \neq = 0, if x = 0$

No, the answer is incorrect.
Score: 0

5. Let $f$ be a polynomial of degree 5, which is given by

$f (x) = a_{5} x^{5} + a_{4} x^{4} + a_{3} x^{3} + a_{2} x^{2} + a_{1} x + a_{0} .$

Let $f^{'} (b)$ denote the derivative of $f$ at $x = b$ . Choose the set of correct options.

No, the answer is incorrect.
Score: 0

6. Let $f$ be a differentiable function at $x = 3$ . The tangent line to the graph of the function $f$ at the point $(3, 0)$ , passes through the point $(5, 4)$ . What will be the value of $f^{'} (3)$ ?

No, the answer is incorrect.
Score: 0

7. Let $f$ and $g$ be two functions which are differentiable at each $x \in R$ . Suppose that, $f (x) = g (x^{2} + 5 x)$ , and $f^{'} (0) = 10$ . Find the value of $g^{'} (0)$ .

No, the answer is incorrect.
Score: 0

8. Consider a function $f : R \to R$ defined as $f (x) = {15 x ^{3} s i n 12 x + A s i n x B if x \neq = 0, if x = 0.$ If $f (x)$ is continuous at $x = 0$ , then find the value of $90 B - A$

No, the answer is incorrect.
Score: 0

9. The distance (in meters) traveled by a car after $t$ minutes is given by the function $d (t) = g (6 t^{3} + 5 t^{2} + 8 t + 5)$ , where $g$ is a differentiable function with domain $R$ . Find the instantaneous speed of the car after 5 min, where $g^{'} (920) = 2$

No, the answer is incorrect.
Score: 0

No, the answer is incorrect.
Score: 0

No, the answer is incorrect.
Score: 0

12. Consider the following function
$q : R \to R q (t) = ∣ t (t - 2) (t - 8) ∣.$

If $m$ is slope of the tangent of the function $q (t)$ at point $t = 2 3$ , find the value $m - 4 27$

No, the answer is incorrect.
Score: 0

No, the answer is incorrect.
Score: 0

No, the answer is incorrect.
Score: 0

No, the answer is incorrect.
Score: 0