2.1 The Tangent and Velocity Problems

Flashcards covering vocabulary, key formulas, and specific worked problems from Section 2.1 on Tangent and Velocity Problems.

Slope formula

$m = \frac{y_2 - y_1}{x_2 - x_1}$, representing the ratio of rise over run.

Slope-intercept formula

$y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Point-slope formula

$y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Tangent line

A line that grazes a curve at a specific point on the function.

Tangent line equation for $f(x) = x^3$ at $(2,8)$

$y = 12x - 16$, calculated using the estimated slope $m = 12$.

Tangent line equation for $y = \cos(\pi x)$ at $P(0.5, 0)$

$y = -\pi x + 0.5\pi$, calculated using the estimated slope $m = -\pi$.

Distance formula

$\text{Distance} = \text{rate} \cdot \text{time}$ ($D = r \cdot t$).

Average velocity formula

$\text{Average Velocity} = \frac{\Delta \text{distance}}{\Delta \text{time}}$.

Delta symbol ($\Delta$)

A symbol representing the change in a variable, such as change in distance or change in time.

Mars rock height equation

$y = 10t - 1.86t^2$, which gives the height in meters $t$ seconds after a rock is thrown on Mars with a velocity of $10\,m/s$.

Average velocity of Mars rock on interval $[1, 2]$

$4.42\,m/s$, calculated as $\frac{12.56\,m - 8.14\,m}{2\,s - 1\,s}$.

Estimated instantaneous velocity of Mars rock at $t = 1$

$6.28\,m/s$, estimated by taking average velocities over progressively smaller intervals near $t = 1$.