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Calculus Level 5

$\left| 3\sin x - 4\cos x + t + \cfrac 4t \right|$

For all real $t$ , let $f(t)$ be the minimum of the above expression over $x\in \mathbb{R}$ .

Let $\displaystyle \int_1^{10} f(t) \, dt = 4 ( a + \ln b )$ for some rational $a$ and $b$ .

Find the value of $a + 2b$ .

The answer is 8.00.

1 solution

Rajen Kapur
Nov 25, 2014

3 sin x - 4 cos x takes values from -5 to 5 as t varies from 1 to 10 in such a way as to make the given expression a minimum. For t = 1 to 4 the expression is 0. For t = 4 to 10 integrate using the value of 3 sin x - 4 cos x = -5, to get a=3 and b = 2.5. Answer=8

So if i well understand your solution, the problem is actually not correctly stated: t is the variable in the integral, and x is the variable from which we calculate the minimum. It should be written "let f(t) be the minimum value of the expression above", such that for t = 1 to 4 this is 0, for t = 4 to 10 this is (-5 + t + 4/t).

mat baluch - 4 years, 7 months ago

Thanks. Those who answered 2 have been marked correct.

I have updated the problem to say:
- For all real $t$ , let $f(t)$ be the minimum of the above expression over $x \in \mathbb{R}$ .
- Find $\int f(t) \, dt$ (for consistency).

Does the problem look good now?

Calvin Lin Staff - 4 years, 7 months ago

Yes it does, thank you !

mat baluch - 4 years, 7 months ago

Think More , Calculate Less !

The answer is 8.00.

1 solution

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