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) = x³; P (1,1)

Accepted Answer

Welcome back, everyone. Use the following limit definition to determine the slope of the line tangent to the graph of F at P where F of X is equal to 5x cubed and P is a point with X equals -1, Y equals -5. The equation is given to us and we're given for answer choices A says -3, B12, C 15, and D 24. So now if we carefully analyze the given equation that says m, which is the slope of the tangent line, is equal to the limit as H approaches 0 of F A plus H minus F of A divided by H. and we have to understand that the point of interest P is defined as -1 -5, and this means that A, which is the X value of the point of interest is -1. And now F of A is the Y coordinate, right? And the Y coordinate is -5. So we already know what FFA is, so all that we have to do is evaluate F at point A plus H, right? And our F of X is given, it's 5 x cubed. So instead of X, we now have to replace that X with A plus H. But we also have to remember that A is -1. So we want to evaluate 5 multiplied by -1 plus H cubed. We have successfully replaced X with -1 plus H, and now we need to simply apply the formula for the sum of two terms cubed. Let's factor out the constant. We have 5. 1st, we have to keep the first term, -1 cubed. Then we have 3 multiplied by the square of the first term, -1 squared multiplied by the second term H plus 3 multiplied by the first term. Multiplied by the square of the second term, so -1 multiplied by H2d. Finally, we add the cube of the second term which is h cubed. And now we can simplify and distribute the terms -1 cubed is -1, 5 multiplied by -1 is -5. Then we have 3H multiplied by 5, which gives us 15H. Then we have -3 H2d multiplied by 5, which is -15 H2, and finally we end up with plus 5 H cubed. Well done. Now we are ready to apply the formula, so the slope equals limit as H approaches 0. And now we have to substitute the whole expression -5 plus 15 minus 15 H2 + 5 H cubed minus F of A. So we're subtracting -5. All of that divided by H. Let's simplify and combine the like terms. So we get -5 minus -5, that's -5 plus 5. We can essentially cancel out those terms, and what we notice is that all of the terms in the numerator contain h. So we can factor out the H, which gives us h. So first of all, age, right? But specifically it will be 15, because if we begin with the first term, 15H, if we factor out H this leaves us with 15 minus 15 H, and then we get Plus 5 H cubed, if we factor out H, we end up with 5 H squared. It is not completely factored, right? We can still factor out the 5, but so far it is not necessary because we just want to cancel out the H. And now we want to substitute h equals 0 and see if we get a finite answer. So we only have the numerator left, which is 15 minus 15 multiplied by 0 + 5 multiplied by 0 squared. So as we can see, this gives us 15, right? There was no necessity to factor out the 5. We have our final answer which corresponds to the answer choice C. 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) = x³; P (1,1)

Key Concepts

Tangent Line

Derivative

Power Rule

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