In Exercises 19–22, find the slope of the curve at the point i...

Question

In Exercises 19–22, find the slope of the curve at the point indicated.

y = x³ − 2x + 7, x = −2

Accepted Answer

Determine the slope of the tangent line to the curve at a specified point. Y equals 2 x cubed minus 3X2 + 1, or X equals 1. We have 4 possible answers, being -1, 01, and 2. Now to solve this, we will make use of the difference quotient. Where we have the limit As H approaches 0. Of F of X0 plus H minus F of X0, all divided by H. In this case, X0 will be 1. We can write our limit definition as limit, as H approaches 0, F1 plus H minus F of 1. Divided by H. Let's go ahead and solve this. So, we have F of 1 plus H first. Which will be 2, multiplied by 1 plus H cubed. -3, multiplied by 1 plus H squared. Plus one. Now, let's simplify this. 1 plus H cubed. While simplify And expand To be one Plus 3 H. Plus 3 H squad, plus HQ. 1 plus H squared also expands. 21 + 2 H plus H squad. We can combine these into our function. To get 2 multiplied by 1 plus 3 H plus 3 H squad, plus HQ. -3, multiplied by 1, plus 2 H plus H squared, all plus 1. Now, we can simplify everything that we have here. We have 2 Plus 6 H plus 6 H squared, plus 2 H cubed. -3 Minus 6 H. Minus 3 H2 + 1. You can keep simplifying, and notice that 6H negative 6H cancels. We have 6H2 minus 3H2, which gives us a 3H2. We have a 2 inch cued, that's by itself, and we have 2 minus 3 plus 1. Which gives us 0. We have 3 H2 plus 2 HQ. For F of 1 plus H. Now, we just need F of 1. F of 1 Will be 2 multiplied by 1 cubed, minus 3 multiplied by 1 squared, plus 1. Which is 2 -3 Plus one. You simplify this, we end up getting F 1 equals 0. Now we'll plug this back into our difference quotient. Limit as H approaches 0. Of F of 1 plus H, which is 3H2 + 2HQ. 0 for F of 1, all divided by H. We have 3 H2d plus 2 H cubed, all divided by H. Now, we noticed then that we can factor out an H in our numerator. We get limit as H4 to 0 of H multiplied by 3 H plus 2 H2, all divided by H. And if we cancel it, we get limit as H approaches 0 of 3H. Plus 2 inch squared. If we then take our limit by plugging in their value, we get 3 multiplied by 0, plus 2 multiplied by 0 squared. Which means the limit, as H approaches 0. It's just 0. That means our slope will be equals. To 0. Now, if we go back to look at our possible answers, we can determine the answer to our problem is answer B. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

In Exercises 19–22, find the slope of the curve at the point indicated.

y = x³ − 2x + 7, x = −2

Key Concepts

Derivative

Slope of a Curve

Evaluating Functions

Watch next