AP Calc Notes: DA2 – 1 Linearization (approximating curves with a model of a line)
Ex: A line passes through the point (2, 5) and has slope 0.42.
a. Write the equation of the line.
b. What is y when x = 2.3?
c. How much does y change when x changes by 0.3?
Note: For any linear function y = mx + b,
Ex: Let f be a differentiable function.
a. Write the equation of the tangent line to f at the point
where x = a. (This is sometimes called the
“linearization” of the function at x = a.)
y
y = f(x)
Point (a, f(a)), slope f'(a)
y = f'(a)(x – a) + f(a) or
y = f(a) + f'(a)(x – a)
a
b. If x = b is a point “near” x = a, use the tangent line to approximate f(b).
x
b
f(b) ≈ f(a) + f'(a)(b – a)
c. The tangent line values lie above the curve (overestimate) if the curve is ________________
The tangent line values lie below the curve (underestimate) if the curve is ________________
Increasing
Concave up
Decreasing
Concave up
y
Increasing
Concave down
y
y
−3
−2
y
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
x
−4
Decreasing
Concave down
−1
1
2
3
4
1
x
5
−4
−3
−2
−1
1
2
3
4
x
5
−4
−3
−2
−1
1
2
3
4
x
5
−4
−3
−2
−1
1
−1
−1
−1
−1
−2
−2
−2
−2
−3
−3
−3
−3
−4
−4
−4
−4
2
3
4
5
Ex: f ( x ) = x
a. Find the linearization of the function at x = 9.
b. Approximate f(8).
c. Is this an overestimate or underestimate? Justify your reason.
Ex: f(x) = sin-1x
a. Find the linearization of the function at x = 0.5.
b. Approximate f(0.45).
c. Is this an overestimate or underestimate? Justify your reason.
Ex: According to the National Weather Service, when the actual temperature is 0° F, the apparent “windchill
temperature” is given by w = 35.74 − 35.75v 0.16 where v is wind velocity in mph.
a. Find the linearization of this function at v = 10.
b. Approximate the windchill temperature when v = 15 mph.
c. Is this an overestimate or underestimate? Justify your reason.