Use linear approximation to show that each function below can be approximated by the given expression when |x| is small
i.) sinx ~ x
ii.) e^x ~ 1+x
iii.) ln(1+x) ~ x
Sure! Let's use linear approximation to show the approximations for each function when |x| is small.
i.) To approximate sin(x) ~ x, we can use the tangent line approximation at x = 0.
The equation of the tangent line to the graph of sin(x) at x = 0 is given by y = f(0) + f'(0) * (x - 0).
Taking the derivative of sin(x) with respect to x, we get f'(x) = cos(x). Evaluating at x = 0, we have f'(0) = cos(0) = 1.
Plugging these values into the equation of the tangent line, we have y = sin(0) + cos(0) * (x - 0). Since sin(0) = 0 and cos(0) = 1, the equation simplifies to y = x. Therefore, sin(x) can be approximated by x when |x| is small.
ii.) To approximate e^x ~ 1 + x, we can use the tangent line approximation at x = 0.
The equation of the tangent line to the graph of e^x at x = 0 is given by y = f(0) + f'(0) * (x - 0).
Taking the derivative of e^x with respect to x, we get f'(x) = e^x. Evaluating at x = 0, we have f'(0) = e^0 = 1.
Plugging these values into the equation of the tangent line, we have y = e^0 + 1 * (x - 0). Since e^0 = 1, the equation simplifies to y = 1 + x. Therefore, e^x can be approximated by 1 + x when |x| is small.
iii.) To approximate ln(1+x) ~ x, we can use the first-order Taylor series expansion.
The first-order Taylor series approximation for a function f(x) = ln(1+x) is given by f(x) ≈ f(a) + f'(a)(x-a), where a is a point near which we want to approximate the function.
Let's choose a = 0. Taking the derivative of f(x) = ln(1+x) with respect to x, we get f'(x) = 1/(1+x). Evaluating at x = 0, we have f'(0) = 1.
Plugging these values into the Taylor series approximation formula, we have f(x) ≈ f(0) + f'(0)(x-0). Since f(0) = ln(1+0) = 0, the equation simplifies to f(x) ≈ 0 + 1(x-0), which is just f(x) ≈ x. Therefore, ln(1+x) can be approximated by x when |x| is small.