Use definition (2) (p. 135) to find the slope of the line tang...

Question

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = √(x + 3); P (1,2)

Accepted Answer

Welcome back, everyone. Use the following limit definitions to determine the slope of the line tensions to the graph of FFP where F of X is equal to square root of X + 7 and P is a point with X of 9 and Y of 4. The slope of the tangent line is equal to the limit as H approaches 0 of F at point A plus H minus F of A divided by H. Were given 4 answer choices A says 1 divided by 8, B 3 divided by 7, C 5 divided by 11, and D says 4. So let's use the definition. It says that the slope of the tangent line is the given limit. So we're just going to rewrite this limit and analyze the terms. We want to identify the function value at a plus H, subtract the function value at a and divide by H. So what is a? And while we have to recall that a is the x coordinate of the point of tangency and here it is 9 and F of A. In this case, it is the Y coordinate of the point of tangency which is 4. So now what we're going to do is substitute. For F of A plus H first. Now this means that for the given X we are replacing it with a plus H in the function, so we get square root of. A plus H. A is 9, right? So we get 9 plus H instead of X, and then we need to add 7 because this is a free coefficient under the radical. Then we subtract F of A, which is 4, and then we divide it by H. If we now combine the like terms, we're going to get the limit as H approaches 0 of square root of H + 16, we are subtracting 4 and dividing by H. The direct substitution will give us 0 divided by 0 form. So what we're going to do is simply Multiply both sides of the given fraction by the conjugate, right? And this means that we want to identify the term that has a radical. That's our numerator and we want to multiply both sides of our fraction by the same term. But with a positive sign, so square root of H + 16 + 4 instead of -4. And divide by the same term to make sure that we're not changing the original fraction, right? From here, we can apply. The formula for the difference of squares factorization in reverse, right? So essentially we have a factored form. If we multiply the two numerators, we're going to get the square of the first one, which gives us H + 16 minus the square of the second term, which is minus 42. So minus 16. And then in the denominator we have H multiplied by square root of H + 16 + 4. Now, 16 minus 16 gives us 0, and this also means that we can cancel out H divided by H, right? The common factor. We can now write the result on limit. Limit SH approaches 0 of 1 divided by square root of H + 16 + 4. And from here we can apply the direct substitution. So we get 1 divided by if H is 0, we get square root of 16, which is 4. And then we're adding another 4. To get 1/8. So the final answer to this problem corresponds to the answer choice A. Thank you for watching.

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = √(x + 3); P (1,2)

Key Concepts

Tangent Line

Derivative

Limit Definition of Derivative

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