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Chapter 1: Q12-1P (page 1)

If $y = \int_{0}^{x} s i n t^{2} d t$ , find $\frac{d y}{d x}$ .

Short Answer

Expert verified

The value of $\frac{d y}{d x}$ is $\frac{\sin x}{2 \sqrt{x}}$ .

Step by step solution

01

Given Information

Given that $y = \int_{0}^{\sqrt{x}} \sin t^{2} d t$ . ...(1)

02

Formula Used

We know that $\frac{d}{d x} \int_{u (x)}^{v (x)} f (x, t) d t = f (x, v) \frac{d v}{d x} - f (x, u) \frac{d u}{d x} + \int_{u}^{v} \frac{\partial f}{\partial x} d t$ ...(2)

03

Find dydx

Compare equation (1) and equation (2).

$f = \sin (t^{2}), u = 0 and v = \sqrt{x}$

Substitute values of f, u and v in the equation (2).

$\frac{d y}{d x} = f (\sqrt{x}) \frac{d \sqrt{x}}{d t} - f (0) \frac{d (0)}{d x} = \sin x \times \frac{1}{2} x^{- \frac{1}{2}} = \frac{\sin x}{2 \sqrt{x}}$

Hence the value of $\frac{d y}{d x}$ is $\frac{\sin x}{2 \sqrt{x}}$ .

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Most popular questions from this chapter

$In \cos x$ Hints:Method1:Write $\cos x = 1 + (\cos x - 1) = 1 + u$ ;use the series you know for $In (1 + u)$ ;replace u by the Maclaurin series for $(\cos x - 1)$

Method2: $In \cos x =$ $- \int_{0}^{x} \tan udu$ Use the series of Example 2 in method B.

Find the interval of convergence of each of the following power series; be sure to investigate the endpoints of the interval in each case $\sum_{n = 1}^{\infty} \frac{(- 1)^{n} x^{n}}{n (n + 1)}$ .

The velocityof electrons from a high energy accelerator is very near the velocityof light. Given the voltage Vof the accelerator, we often want to calculate the ratio v / c. The relativistic formula for this calculation is (approximately, for $V ≫ 1$ )

$\frac{v}{c} = \sqrt{1 - {(\frac{0.511}{V})}^{2}}$ , V=Number of million volts

Use two terms of the binomial series (13.5) to find1 - v/cin terms ofV. Use your result to find 1 - v/cfor the following values of V. Caution: V= the number of millionvolts.

(a) V =100 million volts

(b)V =500 million volts

(c)V =25,000 million volts

(d)V =100 gigavolts (100109 volts105 million volts)

Use Maclaurin series to evaluate each of the following. Although you could do them by computer, you can probably do them in your head faster than you can type them into the computer. So use these to practice quick and skillful use of basic series to make simple calculations.

$\frac{d^{10}}{{dx}^{10}} (x^{8} \tan^{2} x) at x = 0 .$

$f (x) = e^{x}, a = 3 .$

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