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Chapter 4: Q. 51 (page 400)

Find a function fthat has the given derivative f'and value f(c). Find an antiderivative of f'by hand, if possible; if it is not possible to antidifferentiation by hand, use the Second Fundamental Theorem of Calculus to write down an antiderivative.

f鈥�(x)=1x3+1,f(2)=0

Short Answer

Expert verified

Ans: The function is,f(x)=鈭�2x鈥�1t3+1dt

Step by step solution

01

Step 1. Given information.

given,

f鈥�(x)=1x3+1,f(2)=0

02

Step 2. The objective is to find a function f meeting the above values.

Now, if f'is continuous on [a,b]then for allx鈭�[a,b]

ddx鈭�ax鈥�f(t)dt=f(x)

03

Step 3. Find function,

If Fis an antiderivative of fand fis continuous on [a,b]then F(x)=鈭�ax鈥�f(t)dtfor all x鈭�[a,b]

Using the fact that f(2)=0the derivative is,

f(x)=鈭�2x鈥�1t3+1dt

Therefore, the function is f(x)=鈭�2x鈥�1t3+1dt

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Most popular questions from this chapter

Prove that 鈭�13(3x+4)dx=20in three different ways:

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(b) geometrically, by recognizing the region in question as a trapezoid and calculating its area;

(c) with formulas, by using properties and formulas of definite integrals.

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