Last One Calculus Help Plz

Question

Carry out three steps of the Bisection Method for f(x)=3^x-x^4 as follows:

(a) Show that f(x) has a zero in [1,2].

(b) Determine which subinterval, [1,1.5] or [1.5,2], contains a zero.

(c) Determine which interval, [1,1.25], [1.25,1.5], [1.5,1.75], [1.75,2], contains a zero.

In part (b), the interval with a zero is _______

In part (c), the interval with a zero is _______

Answer 1

Carry out three steps of the Bisection Method for f(x)=3^x-x⁴ as follows:

(a) Show that f(x) has a zero in [1,2].

(b) Determine which subinterval, [1,1.5] or [1.5,2], contains a zero.

(c) Determine which interval, [1,1.25], [1.25,1.5], [1.5,1.75], [1.75,2], contains a zero.

Part (a):

Since f(x) is continuous, there is a zero in [a,b] if f(a) and f(b) have differing signs. In this case, let a=1 and b=2. Then f(a) = f(1) = 3¹ - (1)⁴ = 3-1 = 2 >0.

f(b) = f(2) = 3²-2⁴ = 9 - 16 = -7 < 0

So since f(1) and f(2) have different signs, there is a zero in [1,2].

Part (b):

If a_o = 1 and b_o = 1.5,

f(a_o) = f(1) = 3¹ - (1)⁴ = 3 - 1 = 2 > 0

f(b_o) = f(1.5) = 3^1.5-1.5⁴ = 5.2 - 5.1 = 0.1 > 0 so the root does not reside in that interval.

Since we know there is a root in [1,2] and we know that it is not in [1,1.5], then it must be in [1.5,2] by the process of elimination.

Part (c):

[1,1.25], [1.25,1.5], [1.5,1.75], [1.75,2]

From Part (b) we know it's in [1.5,2] so the first and second intervals are eliminated leaving [1.5,1.75], [1.75,2].

Set a₁ = 1.5 and b₁ = 1.75

f(a₁) = f(1.5) = 3^1.5 - 1.5⁴ = 0.13 > 0

f(b₁) = f(1.75) = 3^1.75 - 1.75⁴ = -2.5 < 0

since the signs are different, there is a root between 1.5 and 1.75.

Now just to make sure there are no more roots in [1,2], we check the interval [1.75,2]:

f(1.75) = -2.5 from above.

f(2) = 3²-2⁴ = 9 - 16 = -7

No more roots since these two function values have the same signs.