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4.3 CURVE SKETCING
CURVE SKETCHING

To draw a curve without a calculator, we must
know the following things:
1.
Zeros

2.
3.
4.
Asymptotes
Points of discontinuity
Local mins/local maxes

5.
Set orginal function = 0 and solve
Find critical points by taking derivative, setting = 0, and
solving.
Concavity
FIRST DERIVATIVE TEST

We have already discussed how to determine if a
point is a local min or a local max.
 First
Derivative Test
 At
a critical point, if f’ changes signs, then it is a local
extreme.
 If f’ does not change sign at a critical point, then this point
is not an extreme value.
 At endpoints:
If f’ < 0 after the left endpoint, then the left endpoint is a local max.
 If f’ > 0 after the left endpoint, then the left endpoint is a local min.
 If f’ < 0 before the right endpoint, then the right endpoint is a local
min.
 If f’ > 0 before the right endpoint, then the right endpoint is a local
max.

FIRST DERIVATIVE TEST

Example:
 Use
the First Derivative Test to find the local
extreme values of g(x) = (x2 – 3)ex.
CONCAVITY

Concavity talks about the way a graph is turned.
 “Smiley
face” = concave up
 “Frowny face” = concave down

To find where a graph is concave up, find the
points of inflection
 The
place on a graph where concavity changes.
 Find second derivative and set = 0.
CONCAVITY

Example:
 Find
all points of inflection of the graph of y = e-x2.
CURVE SKETCHING

Example
 Sketch
the curve f(x) = x3 – 5x2 + 3x + 6.
CURVE SKETCHING

Example:
A
particle is moving along the x-axis with position
function x(t) = 2t3 – 14t2 + 22t – 5 for t ≥ 0. Find
the velocity and acceleration and describe the
motion of the particle.
SECOND DERIVATIVE TEST

Another way to determine local mins/local
maxes is by the Second Derivative Test.
 If
at a critical point f’’(c) is negative (concave down),
then f has a local max at x = c.
 If at a critical point f’’(c) is positive (concave up),
then f has a local min at x = c.
 Note: You cannot use this test is f’’(c) = 0 or f’’(c)
fails to exist.
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