a) Find the linear approximation of the function f(x)=4cos(3x^2) near x = 0

b) Use the approximation found in part (a) to estimate the value of f(pi/20)

f'(x) = -24x sin(3x^2)

f'(0) = 0

So, you want the line with slope=0 through (0,4). Duh. y=4

Kind of a dumb place to approximate, eh? With a horizontal tangent, and all that.

a) To find the linear approximation of the function f(x) = 4cos(3x^2) near x = 0, we'll use the formula for linear approximation:

Linear approximation of f(x) = f(a) + f'(a)(x - a)

where f'(a) is the derivative of f(x) evaluated at x = a.

Let's start by finding the derivative of f(x) with respect to x:

f'(x) = -24xsin(3x^2)

Next, we need to evaluate f'(x) at x = 0:

f'(0) = -24(0)sin(3(0)^2) = 0

Now, we can substitute the values into the linear approximation formula:

Linear approximation of f(x) = f(0) + f'(0)(x - 0)

Since we are approximating near x = 0, f(0) is simply f(0) = 4cos(3(0)^2) = 4cos(0) = 4(1) = 4.

So, the linear approximation becomes:

f(x) ≈ 4 + 0(x - 0)
f(x) ≈ 4

b) We can now use the approximation found in part (a) to estimate the value of f(pi/20):

f(pi/20) ≈ 4

Therefore, the estimated value of f(pi/20) using the linear approximation is 4.

To find the linear approximation of a function, we can use the formula:

L(x) = f(a) + f'(a)(x - a)

where L(x) represents the linear approximation of the function f(x), a is the point near which we want to find the approximation, and f'(a) represents the derivative of the function f(x) evaluated at a.

a) Let's start by finding the derivative of the function f(x) = 4cos(3x^2). Using the chain rule, we have:

f'(x) = -24xsin(3x^2)

Now, we need to evaluate f(a) and f'(a) at x = 0:

f(0) = 4cos(3(0)^2) = 4cos(0) = 4(1) = 4

f'(0) = -24(0)sin(3(0)^2) = -24(0)sin(0) = 0

Next, we substitute these values into the formula:

L(x) = f(0) + f'(0)(x - 0)
= 4 + 0(x - 0)
= 4

Therefore, the linear approximation of the function f(x) = 4cos(3x^2) near x = 0 is L(x) = 4.

b) Now, let's use the linear approximation we found in part (a) to estimate the value of f(pi/20). Since the linear approximation is constant, we can use the value of L(x) we obtained, which is 4:

L(pi/20) = 4

Therefore, the estimate for f(pi/20) using the linear approximation is 4.