(a) Give the linear approximation for the function e−x1+x near x=0.
(Enter ∗ for multiplication: type 2*x for 2x. Enter / for division: type 1/2 for 12.)
- sin responder
(b) Give the quadratic approximation for the function ln(cosx) near x=0.
(Enter ∗ for multiplication: type 2*x for 2x. Enter / for division: type 1/2 for 12. Type ∧ for exponents: x∧2 for x2.)
- sin responder
(a) To find the linear approximation for the function e^(-x1+x) near x=0, we can use the first-order Taylor polynomial. The general formula for the first-order Taylor polynomial is given by:
P1(x) = f(a) + f'(a)(x - a)
In this case, f(x) = e^(-x1+x) and we want to approximate it near x=0.
Step 1: Find f(0) and f'(0)
f(0) = e^(-0)(1+0) = 1
f'(x) = d/dx (e^(-x1+x))
= -(1-1)e^(-x1+x)
= -e^0
= -1
f'(0) = -1
Step 2: Calculate P1(x)
P1(x) = f(0) + f'(0)(x - 0)
= 1 + (-1)(x)
= 1 - x
Therefore, the linear approximation for the function e^(-x1+x) near x=0 is 1 - x.
(b) To find the quadratic approximation for the function ln(cosx) near x=0, we can use the second-order Taylor polynomial. The general formula for the second-order Taylor polynomial is given by:
P2(x) = f(a) + f'(a)(x - a) + (1/2)f''(a)(x - a)^2
In this case, f(x) = ln(cosx) and we want to approximate it near x=0.
Step 1: Find f(0), f'(0), and f''(0)
f(0) = ln(cos(0)) = ln(1) = 0
f'(x) = d/dx (ln(cosx))
= -(1/cosx)(-sinx)
= sinx/cosx
= tanx
f'(0) = tan(0) = 0
f''(x) = d/dx (tanx)
= sec^2x
f''(0) = sec^2(0) = 1
Step 2: Calculate P2(x)
P2(x) = f(0) + f'(0)(x - 0) + (1/2)f''(0)(x - 0)^2
= 0 + 0(x) + (1/2)(1)(x^2)
= (1/2)x^2
Therefore, the quadratic approximation for the function ln(cosx) near x=0 is (1/2)x^2.