Explain the concept of local linearity.
Zooming in on a curve until it appears as a straight line; using the tangent line to approximate function values near the point of tangency.
Explain how tangent lines are used for approximation.
The tangent line at a point closely approximates the function's values near that point. Plug in the x-value into tangent line equation to approximate the y-value.
How does the distance from the point of tangency affect accuracy?
Approximations are most accurate close to the point of tangency and become less accurate as you move further away.
Explain the relationship between concavity and over/underestimation.
Concave up means the tangent line is an underestimate. Concave down means the tangent line is an overestimate.
Describe the steps to approximate a function value using linearization.
Find the point of tangency, calculate the derivative at that point, build the tangent line equation, and plug in the x-value to approximate.
Explain why linearization works best for values close to the point of tangency.
The closer you are to the point of tangency, the more closely the tangent line resembles the original function.
Describe the relationship between the derivative and the tangent line.
The derivative of a function at a point is the slope of the tangent line to the function at that point.
How does the sign of the second derivative relate to concavity?
A positive second derivative indicates concave up, while a negative second derivative indicates concave down.
What does the first derivative tell you about the function?
The first derivative tells you about the increasing/decreasing behavior of the function.
Why is differentiability important for linearization?
Differentiability ensures the existence of a tangent line, which is essential for linearization.
How do you find the equation of the tangent line given a function and a point?
1. Find the derivative of the function. 2. Evaluate the derivative at the given x-value to find the slope. 3. Use the point-slope form to write the equation of the line.
How do you approximate $f(x)$ using linearization?
1. Find the tangent line at a nearby point $a$. 2. Plug $x$ into the tangent line equation to find the approximate value.
How do you determine if a linear approximation is an overestimate or underestimate?
1. Find the second derivative of the function. 2. Determine the concavity at the point of tangency. 3. Concave up = underestimate, concave down = overestimate.
How do you find the value of $f(1.1)$ given the tangent line at $x=1$?
1. Substitute $x = 1.1$ into the equation of the tangent line. 2. Solve for y. The y-value is the approximate value of $f(1.1)$.
How do you approximate $f'(3)$ using data from a table?
1. Find two points near $x=3$ in the table. 2. Calculate the slope between those two points using the difference quotient.
How do you find the equation of the tangent line to $f(x)$ at $x=a$?
1. Find $f(a)$. 2. Find $f'(x)$. 3. Find $f'(a)$. 4. Use the point-slope form: $y - f(a) = f'(a)(x - a)$.
How do you use a tangent line approximation to estimate $f(a+h)$?
1. Find the tangent line at $x=a$: $y - f(a) = f'(a)(x - a)$. 2. Substitute $x = a+h$ into the tangent line equation. 3. Solve for $y$.
How do you determine concavity given $f(x)$?
1. Find the second derivative, $f''(x)$. 2. Determine the sign of $f''(x)$. 3. If $f''(x) > 0$, concave up. If $f''(x) < 0$, concave down.
How do you find the slope of the tangent line at $x=a$?
1. Find the derivative, $f'(x)$. 2. Evaluate the derivative at $x=a$: $f'(a)$.
How do you solve for $f(x)$ using the tangent line?
1. Find the equation of the tangent line. 2. Substitute the x-value into the equation of the tangent line. 3. Solve for y.
What is the point-slope form of a line?
$y - y_1 = m(x - x_1)$
What is the linearization formula?
$L(x) = f(a) + f'(a)(x - a)$
How do you calculate the slope, m, for linearization?
$m = f'(a)$, where $a$ is the x-coordinate of the point of tangency.
How to find $f'(x)$ if given $f(x)$?
Differentiate $f(x)$ with respect to $x$.
How to find the equation of tangent line?
Use $y - f(a) = f'(a)(x - a)$, where $a$ is the x-coordinate of the point of tangency.
What is the formula to approximate $f(x)$ using linearization?
$f(x) \approx L(x) = f(a) + f'(a)(x-a)$
How do you find the derivative of a function at a specific point?
Evaluate $f'(x)$ at that point: $f'(a)$.
What is the formula for the second derivative?
The derivative of the first derivative: $f''(x) = \frac{d}{dx} [f'(x)]$.
How do you determine concavity using the second derivative?
If $f''(x) > 0$, concave up; if $f''(x) < 0$, concave down.
What is the formula for approximating f(a+h) using linearization?
$f(a+h) \approx f(a) + h*f'(a)$
