AB Calculus - Hardtke
Assignment 3.14: Linear Approximation
Name _______________________________
Date: Tuesday, Nov 12
Notes on Linear Approximation:
 Sometimes it is not easy to find the y-coordinate at a given x value for some complicated function.
 We notice that in that local area, the points on the tangent line would be a close enough approximation
 So to avoid difficult computations, we use a Linear Approximation (on the tangent line) rather than the
actual value of f(x)
Example 1: Use a linear approximation to estimate the square root of 9.1
Example 2: Find a linear approximation for (2.001)
Differentials: Find
DIFFERENTIAL dy.
Over 
5
dy
2
for y = x . Now multiply both sides by dx. You have just found an expresión for the
dx
1. Sketch a tangent line at each bold point of tangency shown on the graphs below.
A.
B.
C.
D.
2. When is a tangent line BELOW a curve? When is is a tangent line ABOVE a curve?
3. When is a LINEAR APPROXIMATION going to give an overestimate? An underestimate?
4. Find a linear approximation for e
– 0.015
2/3
5. Find a linear approximation for (8.06)
E.
AB Calculus - Hardtke
Assignment 3.14: Linear Approximation
Name _______________________________
Date: Tuesday, Nov 12
Notes on Linear Approximation:
 Sometimes it is not easy to find the y-coordinate at a given x value for some complicated function.
 We notice that in that local area, the points on the tangent line would be a close enough approximation
 So to avoid difficult computations, we use a Linear Approximation (on the tangent line) rather than the
actual value of f(x)
Example 1: Use a linear approximation to estimate the square root of 9.1
Example 2: Find a linear approximation for (2.001)
Differentials: Find
DIFFERENTIAL dy.
Over 
5
dy
2
for y = x . Now multiply both sides by dx. You have just found an expresión for the
dx
1. Sketch a tangent line at each bold point of tangency shown on the graphs below.
A.
B.
C.
D.
2. When is a tangent line BELOW a curve? When is is a tangent line ABOVE a curve?
3. When is a LINEAR APPROXIMATION going to give an overestimate? An underestimate?
4. Find a linear approximation for e
– 0.015
2/3
5. Find a linear approximation for (8.06)
E.