5 Continuous and Differentiable Functions Roller Coaster
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Erin M.
Algebra
1 year, 1 month ago
Develop a piecewise curve with 5 pieces that is continuous and differentiable at all points along the curve and showing a possible roller coaster. You need to show the graph and prove that the function you generate is differentiable (and thus continuous) at the points where the piecewise curves "meet". You may use any type of function we have discussed so far including polynomial, rational, exponential, trigonometric, and others. Try to incorporate at least four different types of functions in your design. Time can be however long you desire as long as there are 5 pieces to the curve that are continuous and differentiable.
Related Question
Use the graph of the function ((x) to sketch graph of f"(x) (5 points) Use the graph of f(x) in the figure to answer the questions: points) The values of at which the function is not continuous islare The values of x at which the function is not differentiable isare:
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Video Transcript
Hi there. In this question we are given the graph of X. And we are being asked find the graph of the secondary with two of the function for the second year. With the function we have to just take care of the shape of the graph. So this is the shape which is a concave up. That means the second derivative of the function in this case is greater than zero. And for the next part um the shape of the graph is concave down. So that means the secondary to of the function is less than zero. And for the other part again the shape of the ffx is um up. That means in this part the secondary with what the function is again greater than zero. So from this um informations we know that. So there is an inflection point. So let's say the inflection is point is somewhere like this one. X one and the other one is X two. Here let's say this is X one is here and X two is here. So before X one, the second derivative of the function has the positive values. That means it is coming from up and down and then going again up. Which means in this part. So in this part the function has the negative values. So the secondary of the function is negative. And for the for the other parts the secondary with the function is negative in this case. And for the other parts, you mean this part and this part the second derivative of the function is greater than zero. The second derivative of the function is greater than zero. So this is the possible graph of this uh f of X function and this is the second derivative of the function graph of the second derivative of the function for the next part of the question use the graph of the, so the values of X at which the function is not continuous. So ford continued to, the function has um has a limit and the value of the function is equal to the limit value. So in this case we are looking for the values so at X is equal to negative one. There's a jump here. So that means there's a jump discontinuity here and there is also um at X is equal to one, we have another discontinuity because this is a removable discontinuity. So we have two points at X is equal to negative one and one we have discontinued. So this is removable discontinue too and this one is jump discontent. And for the next part of the question, the values of X, which the function is not differentiable. So first of all, if the function is not continues, that means the function is not differentiable at those values. So from the part one for X is equal to negative one and X is equal to X is equal to one, the function is not differentiable. Additionally, when you look at the value of X is equal to two, there is a sharp corner. So that means the function is not differentiable because the function is continuous at X is equal to two. But the problem is when you look at the left there with you and the right there with you, they are not equal to each other. That means this function is not differentiable at X is equal to two. Okay, these are the values for this question.
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