Tangent Lines and Derivatives quiz #1 Flashcards
Tangent Lines and Derivatives quiz #1
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What is the main difference between a secant line and a tangent line to a curve?
A secant line intersects a curve at two points, while a tangent line touches the curve at exactly one point. How do you find the slope of the tangent line to the function f(x) = x^2 at x = 1 using limits?
The slope is found by evaluating the limit: lim_{x→1} [(x^2 - 1)/(x - 1)], which simplifies to lim_{x→1} (x + 1) = 2. What is the general formula for the slope of the tangent line (derivative) at a point x = c for a function f(x)?
The slope is given by lim_{x→c} [f(x) - f(c)] / (x - c). Given f(x) = 3x^2 - 4, how do you find the equation of the tangent line at x = -2?
First, find f(-2) = 8. Then, find the slope using the limit definition, which gives -12. Use point-slope form: y - 8 = -12(x + 2), which simplifies to y = -12x - 16. What is the meaning of the derivative at a point in terms of rates of change?
The derivative at a point represents the instantaneous rate of change of the function at that point, or the slope of the tangent line. How do you resolve a 0/0 indeterminate form when finding the slope of a tangent line using limits?
Simplify the numerator, often by factoring, to cancel the denominator and then evaluate the limit. If f(x) = x^2, what is the derivative f'(x) using the limit definition, and what does it represent?
Using the limit definition, f'(x) = lim_{h→0} [(x + h)^2 - x^2]/h = 2x. It represents the slope of the tangent line to f(x) at any point x. What is the main difference between a secant line and a tangent line to a curve?
A secant line intersects a curve at two points, while a tangent line touches the curve at exactly one point. How do you resolve a 0/0 indeterminate form when finding the slope of a tangent line using limits?
You simplify the numerator, often by factoring, to cancel the denominator and then evaluate the limit. What is the general formula for the slope of the tangent line (derivative) at a point x = c for a function f(x)?
The slope is given by lim_{x→c} [f(x) - f(c)] / (x - c).
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