Applications of Differentiation flashcards
36 practice cards drawn from the Applications of Differentiation lessons. Tap a card to turn it over. Every answer is checked against the lesson it came from.
Read the Applications of Differentiation lessons in full →
A tangent has gradient 4. What is the gradient of the normal?
from “Tangents and Normals to Curves”
For at x = 4, the tangent line is
y = 8x − 16
from “Tangents and Normals to Curves”
The linearization of f at a is
f(a) + f'(a)(x − a)
from “The Tangent-Line Approximation”
f(2) = 5 and f'(2) = 3. Estimate f(2.1).
5.3
from “The Tangent-Line Approximation”
on [0, 4]. The value of c the theorem promises is
2
from “The Mean Value Theorem”
The mean value theorem needs f differentiable
on the open interval
from “The Mean Value Theorem”
f'(x) = 0.2 at a point. Is f increasing there?
yes
from “Increasing and Decreasing Functions”
f'(x) = −3 at a point. Is f increasing there?
no
from “Increasing and Decreasing Functions”
. Where is the gradient zero?
x = 2
from “Finding Stationary Points”
. Where is the gradient zero?
x = 3
from “Finding Stationary Points”
f'(2) = 0. Is x = 2 a critical point?
yes
from “Critical Points and Local Extrema”
f has a cusp at x = 1, so f'(1) does not exist. Is x = 1 a critical point?
yes
from “Critical Points and Local Extrema”
At a flat point f'′ = −4. Peak or trough?
peak
from “The Second Derivative Test”
At a flat point f'′ = 6. Peak or trough?
trough
from “The Second Derivative Test”
f' goes +, 0, − across a flat point. Peak or trough?
peak
from “The First Derivative Test”
f'′ = 0 at a flat point. What settles peak or trough?
the sign of f' either side
from “The First Derivative Test”
On [0, 3], f(0) = 3, f(2) = −1 at the one critical point, f(3) = 0. The minimum is
−1
from “Absolute Extrema on a Closed Interval”
The extreme value theorem needs the interval to be
closed
from “Absolute Extrema on a Closed Interval”
f'′ changes from − to + at a point. What is that point?
a point of inflection
from “Points of Inflection”
At a point of inflection, which quantity changes sign?
f'′
from “Points of Inflection”
To find where a curve crosses the x-axis, set
y = 0
from “Curve Sketching with Derivatives”
To tell a peak from a trough, look at
f'′
from “Curve Sketching with Derivatives”
A rectangle has perimeter 24. What side length gives the biggest area?
6
from “Optimization Problems”
A rectangle has perimeter 16. What side length gives the biggest area?
4
from “Optimization Problems”
. What is the velocity?
4t
from “Motion in a Straight Line”
. What is the velocity?
8t
from “Motion in a Straight Line”
, so dA/dr = 2r. At r = 5, r grows 3 per second. dA/dt = ?
30
from “Related Rates”
The balloon link gives while . What is ?
18
from “Related Rates”
x = t, . Eliminate t.
from “Parametric Equations”
, y = 2t. Where is the point at t = 2?
(4, 4)
from “Parametric Equations”
, y = 2t. What is ?
from “Parametric Differentiation”
For a parametric curve, equals
from “Parametric Differentiation”
Why does every term of have coefficient ?
every derivative of at 0 equals 1
from “Maclaurin Series”
The first two terms of the Maclaurin series of sin x are
from “Maclaurin Series”
at 0 goes to
1
from “L'Hôpital's Rule”
By L'Hôpital's rule, at 0 goes to
1
from “L'Hôpital's Rule”