Chapter 10: Q 32 (page 777)

Find a function $f$ that has the given derivative $f^{'}$ and value f(c)
$f^{'} (x) = 2 x s e c x^{2} \tan x^{2}, f (0) = 2$

Short Answer

Expert verified

The anti derivative is $s e c (x^{2}) + 2$

Step by step solution

Given information

The given expression is $f^{'} (x) = 2 x s e c x^{2} \tan x^{2}, f (0) = 2$

Simplification

Given $f^{'} (x) = 2 x s e c x^{2} \tan x^{2}, f (0) = 2$

We will use an intelligent guess-and-check method to find f

Here $s e c x^{2} \tan x^{2}$ is a derivative of $s e c x^{2}$

So our guess can be $s e c x^{2}$

Now, $2 x$ is a derivative of $x^{2}$

Hence we can say that

$f (x) = s e c x^{2}$ $+ c$

Now,

$f (0) = 2$

$\Rightarrow 2 = s e c (0) + c$

localid="1660720299977" $\Rightarrow 2 = 1 + c \Rightarrow c = 2 - 1 \Rightarrow c = 1$

Hence the antiderivative is $s e c^{2} x + 1$

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Find a function $f$ that has the given derivative $f^{'}$ and value f(c)
$f^{'} (x) = 2 x s e c x^{2} \tan x^{2}, f (0) = 2$

Short Answer

Step by step solution

Given information

Simplification

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Find a function $f$ that has the given derivative $f^{'}$ and value f(c)
$f^{'} (x) = 2 x s e c x^{2} \tan x^{2}, f (0) = 2$