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Chapter 4: Q. 45 (page 400)

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below.

d2dx2鈭�23x鈥�t2+1dt

Short Answer

Expert verified

Ans: d2dx2鈭�23x鈥�t2+1dt=54x

Step by step solution

01

Step 1. Given information.

given expression,

d2dx2鈭�23x鈥�t2+1dt

02

Step 2.  The objective is to calculate the derivative.

Now, iffis continuous on[a,b]then for allx鈭�[a,b],

ddx鈭�au(x)鈥�f(t)dt=f(u(x))u鈥�(x)

So,

f(u(t))=(3t)2+1f(u(x))=(3x)2+1u鈥�(x)=3f(u(x))u(x)=33x2+1

03

Step 3. The derivate expression can be written as,

d2dx2鈭�23x鈥�t2+1dt=ddxddx鈭�23x鈥�t2+1dt=ddx3(3x)2+1=3ddx(3x)2+1=3ddx9x2+1=3(18x)+0=54x

Therefore, the answer is54x

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Most popular questions from this chapter

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess- and- check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating. (Hint for Exercise 54: tanx=sinxcosx).

鈭�(tanx+xsec2x)dx.

Use a sentence to describe what the notation鈭�k=2100k means. (Hint: Start with 鈥淭he sum of....鈥�)

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A function f for which the signed area between f and the x-axis on [0, 4] is zero, and a different function g for which the absolute area between g and the x-axis on [0, 4] is zero.

(b) A function f whose signed area on [0, 5] is less than its signed area on [0, 3].

(c) A function f whose average value on [鈭�1, 6] is negative while its average rate of change on the same interval is positive.

Consider the region between f and g on [0, 4] as in the

graph next at the left. (a) Draw the rectangles of the left-

sum approximation for the area of this region, with n = 8.

Then (b) express the area of the region with definite

integrals that do not involve absolute values.

For each function f and interval [a, b] in Exercises 27鈥�33, use the given approximation method to approximate the signed area between the graph of f and the x-axis on [a, b]. Determine whether each of your approximations is likely to be an over-approximation or an under-approximation of the actual area.

f(x)=x2,[a,b]=[0,3]left sum with

a) n = 3 b) n = 6
See all solutions

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