Column B: Tangent equations (a) $ y = -2x $, (b) $ y = x $, (c) $ y = 0 $
Column C: Figure numbers (1, 2, 3)
Solution:
Option 1: (i) â (b) â 3
Function: $ f(x) = x e^x $
Tangent at 0: $ f’(x) = (1 + x)e^x $, $ f’(0) = 1 $, tangent $ y = x $
Figure: 3
Option 2: (ii) â (a) â 1
Function: $ f(x) = e^{-2x} - 1 $
Tangent at 0: $ f’(x) = -2e^{-2x} $, $ f’(0) = -2 $, tangent $ y = -2x $
Figure: 1
Option 3: (iii) â (c) â 2
Function: $ f(x) = e^x - 1 $
Tangent at 0: $ f’(x) = e^x $, $ f’(0) = 1 $, but the tangent is not given as $ y = x $ here; the solution suggests $ y = 0 $ is incorrect for this function, but as per the matching, (iii) â (c) â 2 is given.
Option 4: (iii) â (b) â 3
Not correct as per function and tangent.
Option 5: (ii) â (a) â 1
Repeat of correct option above.
Option 6: (i) â (a) â 1
Incorrect.
Summary:
Correct matches are:
(i) $ x e^x $ â (b) $ y = x $ â 3
(ii) $ e^{-2x} - 1 $ â (a) $ y = -2x $ â 1
(iii) $ e^x - 1 $ â (c) $ y = 0 $ â 2 (though this is not correct mathematically, it is the match given in the solution for the assignment)
Question 2 (Continuity and Limits)
Statement:
Consider $ f(x) = e^x - 1 $ and $ g(x) = x $.
Options:
**(f + g)(x) = e^x - 1 + x$$
Is (f + g)(x) continuous at $ x = 0 $?
Solution:
LHL: $\lim_{x \to 0^-} (e^x - 1 + x) = 0$
RHL: $\lim_{x \to 0^+} (e^x - 1 + x) = 0$
Value at 0: $(f + g)(0) = 0$
Conclusion: $(f + g)(x)$ is continuous at $ x = 0 $1.
Question 3 (Limit of Product)
Statement:
Find $\lim_{x \to 0} f(x)g(x)$, where $ f(x) = e^x - 1 $, $ g(x) = x $.
Solution:
$\lim_{x \to 0} (e^x - 1)(x) = 0$
Answer: $0$
Numerical Answer Type (NAT)
Question 1
Statement:
Let $ f $ be a differentiable function at $ x = 1 $. The tangent line to the curve represented by the function $ f $ at the point $(1, 1)$ passes through the point $(2, 2)$. What will be the value of $ f’(1) $?
Statement:
Let $ f: \mathbb{R} \to \mathbb{R} $ be defined by $ f(x) = x^2 $ and $ g: \mathbb{R} \to \mathbb{R} $ be defined by $ g(x) = x - 5 $. Find the value of $(fg)’(1) - (f \circ g)’(1)$, where $ fg(x) = f(x)g(x) $ and $ f \circ g(x) = f(g(x)) $.
Note:
The above is based on the available content from the Week 8 Practice Assignment Solution PDF1.
If your PDF contains additional or different questions, please upload or specify further details.
The second source2 refers to a Computational Thinking assignment, not the mathematics PDF, and is not included in this summary.
