Tangent Line Approximation (or Linear Approximation) for AP Calculus
Practice problems for these concepts can be found at: More Applications of Derivatives Practice Problems for AP Calculus
Tangent Line Approximation (or Linear Approximation)
An equation of the tangent line to a curve at the point (a, f (a)) is:
y = f'(a)+ f'(a)(x – a) providing that f is differentiable at a. See Figure 9.2-1.
Since the curve of f (x ) and the tangent line are close to each other for points near x = a, f (x ) ≈ f (a)+ f'(a)(x – a).
Write an equation of the tangent line to f (x ) = x3 at (2, 8). Use the tangent line to find the approximate values of f (1.9) and f (2.01).
Differentiate f (x ): f'(x )=3x2; f'(2)=3(2)2 =12. Since f is differentiable at x =2, an equation of the tangent at x =2 is:
- y = f (2)+ f'(2)(x – 2)
- y =(2)3 +12(x – 2)=8+12x – 24=12x – 16
- f(1.9) ≈ 12(1.9) – 16 = 6.8
- f (2.01) ≈ 12(2.01) – 16 = 8.12. (See Figure 9.2-2.)
If f is a differentiable function and f (2)=6 and , find the approximate value of f (2.1).
Using tangent line approximation, you have
- f(2) = 6 the point of tangency is (2, 6);
- the slope of the tangent at x =2 is
- the equation of the tangent is ;
- thus, .
The slope of a function at any point (x, y) is . The point (3, 2) is on the graph of f. (a) Write an equation of the line tangent to the graph of f at x =3. (b) Use the tangent line in part (a) to approximate f (3.1).
- Let y = f(x ), then
- f (3.1) ≈ –2(3.1)+8 ≈ 1.8
Equation of tangent: y – 2= – 2(x – 3) or y = – 2x +8.
Estimating the nth Root of a Number
Another way of expressing the tangent line approximation is:
f(a + Δx ≈ f(a) + f '(a Δ x; where Δ x is a relatively small value.
Find the approximation value of using linear approximation.
Using f(a + Δx ≈ f(a) + f '(a Δ x, let Δ x = 1.
Find the approximate value of using linear approximation.
Let , a = 64, Δx = – 2. Since and , you can use f(a + Δx ≈ f(a) + f '(a Δ x. Thus, .
Estimating the Value of a Trigonometric Function of an Angle
Approximate the value of sin 31°.
Note: You must express the angle measurement in radians before applying linear approximations. 30° = radians and 1° = radians.
Let f (x) = sin x, a = and Δx = .
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