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Find an approximate solution of $\sqrt{x}-\sqrt[3]{x-1}=0.975 .$ Consider the function $f(x)=\sqrt{x}-\sqrt[3]{x-1}-0.975.$(a) show that the given function is continuous for all $x \geq 0, f(8)<0$ and $f(9)>0 .$ Conclude that there is a number $r, 8<r<9,$ such that $f(r)=0$(b) Choose $x_{0}=9,$ and find $T(x).$(c) Instead of solving $f(x)=0,$ use the approximation $f(x) \approx T(x)$ and instead solve $T(x)=0$ to approximate $r.$

(b) $T(x)=\frac{1}{12}(x-9)+\frac{1}{40}$(c) 8.7

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 6

Linearization and Differentials

Derivatives

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04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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Hi in this video we are going to identify all the points where the function F is continues. The function F is defined as the square root of nine minus X. Over square root of x minus six. Um We can call the numerator the function G o fix Z effects and the denominator as age effects. Now ge affects is square Rudolph nine minus X and H of X is square root of x minus nine. Um The function F is the rest of these two functions. The function G is continuous. It's the square root of something and it is continuous at all. X. Where let's just create a bit of space at all. X where nine minus X is greater or equal than zero, which means at all. Ex uh Where uh let's see X is less than or equal to uh nine. So if I have here the real line And I put here nine everywhere here. Right from minus infinity to nine. The enumerate Tory's continues for the denominator we have that it is continuous at all X. Where X -6 is greater or equal than zero, which means that X is greater or equal than six. And now if we take them the realigned we have here six X has to be greater or equal than six. So the denominator is continues on this interval six to plus infinity. Over here, the intersection of these two is actually let's make it a yellow. It's from here to here. This is the intersection of these two. Um intervals. Uh So the intersection of this interval in this interval is the closed interval uh Six? Nine. So the function F is probably going to be continuous on the closed interval six? Nine. However, we have to make sure that the function is also defined. Uh F So that means we don't have zero. The denominator, which means that we cannot includes the 00.6. And in the end our answer is that okay, F is continuous um at all X in the open interval six. Close interval nine. And um that's it. So to summarize um both of the numerator and the denominator are continuous on the closed interval 69, but we cannot have six because six makes the denominator here equal to zero and it has to be excluded. So the function F is continuous on the open interval six closed interval. Close nine. And that's it. We're done.

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