AP Calculus AB
Day 13
Section 3.9
3/23/2016
Perkins
Linear Approximation
Non-calculator application of the tangent line.
Used to estimate values of f(x) at ‘difficult’ x-values.
(ex: 1.03, 2.99, 7.01)
Steps:
a. Find the equation of the tangent line to f(x) at an ‘easy’
value nearby.
b. Plug the ‘difficult’ x-value in to get a reasonable estimate
of what the actual y-value will be.
1. Find the equation of the tangent line to f(x) at x = 1.
y  1  6  x  1
f  x   x 3  3x  5
y  6x  7
6
4
2
1
-2
-4
-6
f 1  1
2
f '  x   3x 2  3
f ' 1  6
2. Use the equation of the tangent line to f(x) at x = 1
to estimate f(1.01).
f  x   x 3  3x  5
6
y  6x  7
y (1.01)  6(1.01)  7
 0.94
4
2
1
2
-2
-4
f 1.01  .939699
-6
This estimate will be accurate as long as the x-value is very close to
the point of tangency.
AP Calculus AB
Day 13
Section 3.9
3/23/2016
Perkins
Linear Approximation
1. Find the equation of the tangent line to f(x) at x = 1.
f  x   x 3  3x  5
6
4
2
1
-2
-4
-6
2
2. Use the equation of the tangent line to f(x) at x = 1
to estimate f(1.01).
f  x   x 3  3x  5
6
4
2
1
-2
-4
-6
2
Finding Differentials
y  f x
Differential
dy
 f 'x
dx
dy  f '  x  dx
 
Change
in y.
To estimate a y-value using a differential:
1. Find a y-value at a nearby x-value.
2. Add the value of your differential.
Slope of tangent
line at a given x.
y  3 sin x
 
Change
in x.
y  x 5
3. Estimate f(0.03) without your calculator.
dy
 3cos x
f 0  0
dx
dy   3cos0 0.03 
dy   3  0.03 
dy  0.09
f  0.03  0  0.09  0.09
4. Estimate f(8.96) without your calculator.
dy
1

dx 2 x  5
 1
dy     .04 
4
dy  0.01
f 9  2
1
f ' 9 
4
f  8.96  2  0.01  1.99
Finding Differentials
y  f x
y  3 sin x
3. Estimate f(0.03) without your calculator.
y  x 5
4. Estimate f(8.96) without your calculator.