ISI2018-MMA-29

Question

Let $f$ be a continuous function with $f(1) = 1$. Define $$F(t)=\int_{t}^{t^2}f(x)dx$$.

The value of $F’(1)$ is

• A. $-2$
• B. $-1$
• C. $1$
• D. $2$

Comments

I cannot remember now.

But I think $f(x)=e^{x}$

Now, integration putting upper bound $\int e^{t^{2}}dt$  is it not form of laplace?

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Its just differentiation with respect to the limit of integration. 

https://math.stackexchange.com/questions/984111/differentiating-with-respect-to-the-limit-of-integration

So I guess answer is C.

 

 

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F′(t)=f(t^2)∗2t -f(t)*1  

Now, put t=1

⇒ F′(1) = f(1)∗2 – f(1)  

⇒F′(1) = 1∗2 -1

So, F′(1) = 2-1 = 1 (Option C).

Answer 1

simply apply Leibnitz and u will get and as 1