Show all of your work, even though the question may not explicitly remind you to do so. Clearly label any functions, graphs, tables, or other objects that you use. Justifications require that you give mathematical reasons, and that you verify the needed conditions under which relevant theorems, properties, definitions, or tests are applied. Your work will be scored on the correctness and completeness of your methods as well as your answers. Answers without supporting work will usually not receive credit.
Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If your answer is given as a decimal approximation, it should be correct to three places after the decimal point.
Unless otherwise specified, the domain of a function fis assumed to be the set of all real numbers for which f() is a real number.
Find the average rate of change of f over the interval -15:34. For how many values of in the interval -15#54 does the instantaneous rate of change of f equal the average rate of change of f over that interval? (b) Write an equation for the line tangent to the graph off at # 1 Please respond on separate paper, following directions from your teacher. (c) exist For each of lim” 16″) and lim 1 – find the value or give a reason why it does not Please respond on separate paper, following directions from your teacher. (d) Let 9 be the function defined by 9(+) = f(+). Find (0) Please respond on separate paper, following directions from your teacher.
The Correct Answer and Explanation is :
It looks like you’re working through a set of problems that involve functions, derivatives, and limits. Let’s break this down step-by-step, even though I can’t directly view the image you provided.
Step-by-Step Approach
Part (a): Average Rate of Change of ff over the Interval [−15,34][-15, 34]
The average rate of change of a function f(x)f(x) over an interval [a,b][a, b] is given by: Average Rate of Change=f(b)−f(a)b−a\text{Average Rate of Change} = \frac{f(b) – f(a)}{b – a}
For your interval [−15,34][-15, 34], we can substitute these values into the formula. If we know the values of f(−15)f(-15) and f(34)f(34), we can calculate the average rate of change over this interval.
Once the average rate of change is calculated, we can move on to finding when the instantaneous rate of change is equal to the average rate of change.
Instantaneous Rate of Change: This refers to the derivative of the function, and the Mean Value Theorem tells us that for continuous and differentiable functions, there is at least one value cc in the interval [a,b][a, b] such that the instantaneous rate of change at that point equals the average rate of change over the entire interval.
Thus, after finding the average rate of change, you’d look for the value(s) of xx in the interval where the derivative f′(x)f'(x) equals the average rate of change.
Part (b): Equation for the Tangent Line at x=1x = 1
To write the equation of the tangent line to the graph of ff at x=1x = 1, we use the point-slope form of the equation of a line: y−f(1)=f′(1)(x−1)y – f(1) = f'(1)(x – 1)
You need to find f(1)f(1) and f′(1)f'(1). If you have the function f(x)f(x), substitute x=1x = 1 into the function to get f(1)f(1), and then take the derivative of the function f′(x)f'(x) and evaluate it at x=1x = 1 to find f′(1)f'(1).
Part (c): Limits of limx→16f(x)\lim_{x \to 16} f(x) and limx→1−f(x)\lim_{x \to 1^-} f(x)
To find these limits, you must evaluate the behavior of the function as xx approaches 16 from the left and right (for the first limit) and as xx approaches 1 from the left for the second limit.
- If the function is continuous at those points, you can directly substitute the values of xx into the function.
- If the function is not continuous, then you need to apply limit laws or check for one-sided limits.
If the function doesn’t approach a specific value or behaves erratically (like having an asymptote), the limit does not exist.
Part (d): Finding g(0)g(0) for the Function g(x)=f(x)g(x) = f(x)
This part is asking for the value of g(0)g(0). Since g(x)=f(x)g(x) = f(x), you simply need to evaluate f(0)f(0). g(0)=f(0)g(0) = f(0)
Explanation:
You will follow these steps for each of the parts in the problem. For each, make sure to use the relevant formulas and verify the conditions under which the functions or theorems are valid. Use precise notation, and if you encounter any specific challenges in calculating, break the problem into manageable pieces.