SME
Calculus I · Problem type
Linear approximation of √x
Use this for “Use the linearization of √x at a = 25 to estimate √26.”
Chart
Tangent at x = 4: slope f′(4) = 0.25, y = 0.25x + 1
Linear approximation L(4.1) = 2.025; the true f(4.1) = 2.0248, off by 0.0001543
This is an example. Type your own values to start a new problem.
L(x)
=
f(a)
+
f′(a)
(x
−
a
)
x
≈
L(
x
)
=
f(x)
True value √x
calculated
E
Error L(x) − f(x)
worked out (not typed)
Formulas
f
(
a
)
=
√
a
2
=
√
4
f
′(
a
)
=
1
2 ×
f
(
a
)
0.25
=
1
2 × 2
L(
x
)
=
f
(
a
)
+
f
′(
a
)
×
(
x
−
a
)
2.025 = 2 + 0.25 × (4.1 − 4)
f
(
x
)
=
√
x
2.0248457
=
√
4.1
E
=
L(
x
)
−
f
(
x
)
0.0001543 = 2.025 − 2.0248457
Assumptions
•
L(x) = f(a) + f′(a)(x − a) is the tangent line at a. Near a it stays close to the curve, so f(x) ≈ L(x).
•
For f(x) = √x the power rule gives f′(x) = 1 ÷ (2√x), so a must be above 0.
•
Pick an a near x whose square root is known (a perfect square), so f(a) and f′(a) are exact.
•
√x bends down, so its tangent lies above it: L(x) is a little too big, and the error L(x) − f(x) is never negative.
Step-by-step
WE KNOW
a = 4, x = 4.1
FIND
value at a (f(a)), slope at a (f′(a)), linear approximation (L(x)), true value √x (f(x)), error L(x) − f(x) (E)
Step 1 · Find value at a (f(a))
Use f(a) = √a
The value of the curve at the point of tangency.
f
(
a
)
=
√
4
f
(
a
)
=
2
Step 2 · Find slope at a (f′(a))
Use f′(a) = 1 ÷ (2 × f(a))
Power
rule
on
x
1
2
:
bring
1
2
down
and
lower
the
power
to
−
1
2
,
so
f
′(
x
)
=
1
2√
x
.
Use
√
a
=
f
(
a
).
f
′(
a
)
=
1
2 × 2
y(
x
)
=
√
x
→
dy/dx
=
1
2√
x
f
′(
a
)
=
0.25
Step 3 · Find linear approximation (L(x))
Use L(x) = f(a) + f′(a) × (x − a)
The
tangent
line
at
a
is
L(
x
)
=
2
+
0.25(
x
−
4).
Put
in
x
=
4.1.
L(
x
)
=
2
+
0.25
×
(4.1
−
4)
L(
x
)
=
2
+
0.25
×
0.1
L(
x
)
=
2
+
0.025
L(
x
)
=
2.025
Step 4 · Find true value √x (f(x))
Use f(x) = √x
The true value, worked out to compare.
f
(
x
)
=
√
4.1
f
(
x
)
=
2.0248457
Step 5 · Find error L(x) − f(x): E
Use E = L(x) − f(x)
How
far
the
tangent
sits
above
the
curve
at
x
.
E
=
2.025
−
2.0248457
E
=
0.0001543
Check
Put the numbers back in. Both sides should match.
2
=
√
4
✓
0.25
=
1
2 × 2
✓
2.025 = 2 + 0.25 × (4.1 − 4) ✓
2.0248457
=
√
4.1
✓
0.0001543 = 2.025 − 2.0248457 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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