Chapter 2: Q 29 (page 238)

Find
a function $f$ that has the given derivative $f$ $'$ and value f(c).
$f' (x) = x (3 x + 1), f (2) = 4$

Short Answer

Expert verified

Antiderivative is $f (x) = x^{3} + \frac{x^{2}}{2} - 6$

Step by step solution

Step 1:Given information

The given expression is $f' (x) = x (3 x + 1)$ $, f (2) = 4$

Step 2:Simplification

given

$f' (x) = x (3 x + 1) = 3 x^{2} + x$

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

Here $3 x^{2}$ is derivative of $x^{3}$ and x is derivative of $\frac{x^{2}}{2}$

hence,

$f (x) = x^{3} + \frac{x^{2}}{2} + c$

Now, $f (2) = 4$

$\Rightarrow 4 = {(2)}^{3} + \frac{2^{2}}{2} + c$

$\Rightarrow 4 = 8 + 2 + c \Rightarrow c = - 6$

Therefore

$f (x) = x^{3} + \frac{x^{2}}{2} - 6$

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Find
a function $f$ that has the given derivative $f$ $'$ and value f(c).
$f' (x) = x (3 x + 1), f (2) = 4$

Short Answer

Step by step solution

Step 1:Given information

Step 2:Simplification

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Finda function fthat has the given derivative f' and value f(c).f'x=x3x+1,f2=4

Short Answer

Step by step solution

Step 1:Given information

Step 2:Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

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Pure Maths

Logic and Functions

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Study anywhere. Anytime. Across all devices.

Find
a function $f$ that has the given derivative $f$ $'$ and value f(c).
$f' (x) = x (3 x + 1), f (2) = 4$