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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?
$f(a+h) \approx f(a) + h*f'(a)$
What does a tangent line lying below the curve indicate?
The function is concave up, and the tangent line approximation is an underestimate.
What does a tangent line lying above the curve indicate?
The function is concave down, and the tangent line approximation is an overestimate.
How can you visually determine concavity from a graph?
If the graph curves upward, it's concave up. If it curves downward, it's concave down.
How does the slope of the tangent line relate to the derivative graph?
The slope of the tangent line at a point on the original function's graph is the y-value of the derivative graph at that point.
What does the graph of $f'(x)$ tell you about the slope of $f(x)$?
The y-value of $f'(x)$ at any point $x$ gives the slope of the tangent line to $f(x)$ at that point.
How does the concavity of $f(x)$ relate to the graph of $f''(x)$?
If $f''(x) > 0$, $f(x)$ is concave up. If $f''(x) < 0$, $f(x)$ is concave down.
What does a horizontal tangent line on the graph of $f(x)$ indicate about $f'(x)$?
$f'(x) = 0$ at that point.
How can you visually determine if a function is increasing or decreasing from its graph?
If the graph goes up from left to right, the function is increasing. If it goes down, it's decreasing.
How can you identify points of inflection on a graph?
Points where the concavity changes (from concave up to concave down, or vice versa).
What does the area under the curve of $f'(x)$ represent?
The change in the value of $f(x)$.
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.