Local Linearity

Suppose f(x) = x³ – x.

Use a linear approximation at x = 2 to estimate f(2.5).

• 10.5

• 11

• 11.5

• 12

• 4.6

• 4.7

• 4.8

• 4.9

Suppose f (x) = 3x² is graphed in the x-y plane. Will any linear approximations to f overestimate or underestimate the actual function values?

• overestimate

• underestimate

f(4) = 1
f'(4) = 3
Using a linear approximation centered at 4, f(4.5) ≈

• 5/2
• 3/2
• 4/3
• 7/3

f(x) = x²(x+1)²
Which of these is the linear approximation to f centered at x = 1?

• L(x) = 12x - 1
• L(x) = 6x + 5
• L(x) = 12x - 8
• L(x) = 12x - 16

The slope of the line tangent to the graph y = cos(x) at the point x = π/6 is –1/2. What is the linear approximation of cos(x) at π/6 ?

• L(x) = –1/2(x – π/6) + √(2)/2

• L(x) = –1/2(x – π/6) + 1/2

• L(x) = –1/2(x – π/6) + √(3)/2

• L(x) = –1/2(x – π/6) – 1/2

f(x) = x^-2/5
Use a linear approximation centered at x = 32 to approximate (30)^-2/5.

• 17/180
• 25/64
• 23/128
• 41/160

Which of the following represents the linear approximation to f (x) at the point (c, f(c)) ?

• L(x) = f'(c) + f(c)(x - c)

• L(x) = f(c) + f'(c)(x - c)

• L(x) = f'(c) + f(c)(x + c)

• L(x) = f(c) - f'(c)(x - c)

Which of the following does not necessarily represent a derivative?

• Slope of a tangent line to a curve

• A function derived from another function

• The instantaneous rate of change of a function

All the following are steps necessary for writing equation of the tangent line to a curve at the point (a, f(a)) except:

• Find the value of f (a)

• Evaluate the second derivative to obtain f''(a)

• Write equation of the tangent line in point slope form:

  T (x) – f (a) = f ' (a) (x – a)

• Evaluate f ' (a) to obtain the slope of the tangent line