Derivatives as Functions quiz #1 Flashcards
Derivatives as Functions quiz #1
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What does the derivative of a function represent?
The derivative represents the slope of the tangent line to the function at a specific point. What is the limit definition of the derivative for a function f(x)?
The limit definition is f'(x) = lim (h -> 0) [(f(x + h) - f(x)) / h]. How do you find the derivative of f(x) = x² using the limit definition?
Expand (x + h)², subtract x², divide by h, simplify, and take the limit as h approaches 0. What is the general derivative of f(x) = x²?
The general derivative is f'(x) = 2x. How do you find the slope of the tangent line to f(x) = x² at x = 1?
Substitute x = 1 into the derivative: f'(1) = 2 × 1 = 2. How do you find the slope of the tangent line to f(x) = x² at x = -2?
Substitute x = -2 into the derivative: f'(-2) = 2 × (-2) = -4. Why is the limit as h approaches 0 used in the definition of the derivative?
It ensures the two points used to calculate the slope get infinitely close, giving the exact slope at a single point. What does the notation f'(x) mean?
It means the derivative of the function f with respect to x. What is the purpose of finding a general derivative formula for a function?
It allows you to find the slope of the tangent line at any x value without repeating the entire process. What happens if you try to substitute h = 0 too early in the limit definition?
You get division by zero, which is undefined, so you must simplify first. How do you simplify (x + h)² - x²?
Expand to x² + 2xh + h², then subtract x² to get 2xh + h². After simplifying (x + h)² - x², what is the next step in finding the derivative?
Factor h from the numerator and cancel with the denominator. What is the result after canceling h in the derivative calculation for f(x) = x²?
You get 2x + h. What do you do after simplifying to 2x + h in the derivative process?
Take the limit as h approaches 0, resulting in 2x. How can you use the general derivative to find the slope at x = 100?
Plug x = 100 into the derivative: f'(100) = 2 × 100 = 200. What is the benefit of having the general derivative formula?
It saves time by allowing you to quickly find the slope at any point. If f(x) = x², what is the slope of the tangent line at x = 0?
f'(0) = 2 × 0 = 0. What is the meaning of the variable h in the limit definition of the derivative?
h represents the difference between two x-values that approach zero. What is the derivative of f(x) = x² at x = -5?
f'(-5) = 2 × (-5) = -10. How does the derivative relate to the graph of a function?
The derivative at a point gives the slope of the tangent line to the graph at that point. What is the process for finding the derivative of any function using the limit definition?
Substitute f(x + h) and f(x) into the formula, simplify, cancel h, and take the limit as h approaches 0. Why is it important to expand and simplify before taking the limit in the derivative definition?
To eliminate h from the denominator and avoid division by zero. What does it mean if the derivative at a point is negative?
The tangent line at that point slopes downward. What does it mean if the derivative at a point is positive?
The tangent line at that point slopes upward. What is the derivative of f(x) = x² at x = 3?
f'(3) = 2 × 3 = 6. How can you check your derivative calculation for f(x) = x²?
By plugging in specific x-values and comparing with the slope found by other methods. What is the general process for finding the slope of a tangent line to a function at a point?
Find the derivative and substitute the x-value of the point. What is the derivative of f(x) = x² at x = 10?
f'(10) = 2 × 10 = 20. If the derivative at a point is zero, what does this indicate about the tangent line?
The tangent line is horizontal at that point. What is the main advantage of using the general derivative formula over the limit definition each time?
It allows you to quickly find the slope at any point without repeating the limit process. How does the derivative help in understanding the behavior of a function?
It shows how the function changes at each point, indicating increasing or decreasing trends.
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