45–46. Linear approximation

a. Find the linea...

Question

45–46. Linear approximation

a. Find the linear approximation to f at the given point a.

b. Use your answer from part (a) to estimate the given function value. Does your approximation underestimate or overestimate the exact function value?

ƒ(x) = x²⸍³ ; a =27; ƒ(29)

Accepted Answer

Welcome back, everyone. Given the function H X equals x the power of 4/3 at A equals 64, find the linear approximation L of X and use it to estimate L at 66. So let's recall the expression of a linear approximation. L of X is equal to in this case we're given a function H of X, so we're going to use the derivative of H of X. Specifically, were given a point A. So we're going to use H at the point A multiplied by x minus A, right, where A is a point of interest, and then we are adding H at point A, the value of the function at that point. So now what we have to do is differentiate H of X because our first term has the derivative at point A and that's the derivative of X to the power of 4/3. Using the power rule, we're getting 4/3 X to the power of 1/3. And now let's evaluate H at A. Our A is 64, so we're evaluating H at 64. This gives us 4/3 multiplied by 64 rates to the power of 1/3, basically cubic root of 64, right? And cubic root of 64 is 4, so we get 16 divided by 3. Now, what is H of A? Well, it's H at a.64, so that's 64, raises the power of 4/3, right? So, first of all, we can take the cube root of 64, which is 4. And then 4 raises the power of 4 gives us 256. So now our linear approximation L of X equals H at 64, which is 16/3 multiplied by X minus 64 plus H at 64. So we're adding 256. Now let's expand. We're going to get 16/3X minus 16 multiplied by 64 divided by 3. Plus 256, right? So what we're going to do is simply use a common denominator, so that's 256 multiplied by 3 divided by 3. Now let's go ahead and simplify. L of X is equal to 16/3 X. Now what we can do is simply write minus. Use the common denominator of 3, and now let's notice that 256 is simply 16 squad, right? So we can factor out 16. We're going to get 16 and 64, and now we're going to subtract 16 multiplied by 3. Why are we subtracting? Well, that's because we have a negative sign in front and we are adding the final fraction. Now, it's 256 is 16 multiplied by 16, so we have 116 outside, 116 inside, and then We have a multiplication by 3. So this allows us to simplify more easily, and we're going to get L of X equals 16/3X minus. 16 multiplied by 64 minus 48, right? So this gives us 16. And all of that is divided by 3. Now 16 squared is 256. And this is how we get our linear approximation at the given point. So this is our first answer, right? And if we want to use it. To estimate L at 66, so L at 66 is 16 divided by 3. Multiplied by Act, which is 66. And we are subtracting 256 divided by 3. So L at 66 is equal to 16 multiplied by 22, right? Because 66 divided by 3 is equal to. 22. However, if we decide to use a common denominator because we will still need to combine the like terms, we can say that in the numerator we get 16 multiplied by 66 minus 256 while our denominator remains 3. If we perform the calculations in the numerator, we're going to get 800. And our denominator is we, so that's the final answer for L of 66, and we can say that L of 66 is equal to 800 divided by 3. So those will be our final answers to this problem. Thank you for watching.

45–46. Linear approximation

a. Find the linear approximation to f at the given point a.
b. Use your answer from part (a) to estimate the given function value. Does your approximation underestimate or overestimate the exact function value?

ƒ(x) = x²⸍³ ; a =27; ƒ(29)

Key Concepts

Linear Approximation

Derivative

Estimation and Error Analysis

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