Second Derivatives and Beyond Exercises

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f ²(x) is the square of f (x), so we need to square the original function.

Answer

f ⁽²⁾(x) is the second derivative of f (x). The first derivative is f ⁽¹⁾(x) = f '(x) = 3x². The second derivative is f ⁽²⁾(x) = f "(x) = 6x.

Answer

Suppose f (x) is a degree 3 polynomial.

a. The degree of f '(x) is 2, because when we take the derivative the degree decreases by 1.

b. The degree of f ⁽²⁾(x) is 1, because when we take another derivative the degree decreases by 1 again.

c. The degree of f ⁽³⁾(x) is 0. When we've taken as many derivatives as the degree of the original polynomial, we're left with a constant, which has degree 0.

Answer

Suppose f (x) is a degree 9 polynomial.

a. The degree of f ⁸(x) is 1. Each time we take the derivative of a polynomial we lower the degree by 1. After taking 8 derivatives of a degree 9 polynomial, we have only 1 degree left.

b. The degree of f ⁹(x) is 0. After taking 9 derivatives of a degree 9 polynomial, there are no x terms left.

c. The degree of f ¹⁰(x) is 0. Since f ⁹(x) is degree 0 (in other words, a constant), taking another derivative still leaves us with something that's degree 0.

Answer

Suppose f (x) is a degree n polynomial.

a. The degree of f ^{(n – 1)}(x) is 1. After taking (n – 1) derivatives of a degree n polynomial, we're left with something of degree 1.

b. The degree of f ⁽ⁿ⁾(x) is 0. After taking n derivatives of a degree n polynomial we're left with a constant, which has degree 0.

c. The degree of f ^(k)(x) , where k is any integer greater than n, is 0. Since f ⁽ⁿ⁾ is a constant (in other words, has degree 0), any subsequent derivative will be the derivative of a constant, and therefore zero.

Answer

The derivatives of f (x) = sin x follow a pattern. Every 4 derivatives, we find the original function. f (x) is the same as f ⁽⁴⁾(x) is the same as f ⁸(x).

Similarly, f ⁽¹⁾(x) is the same as f ⁽⁵⁾(x) is the same as f ⁹(x) .

If two numbers differ by a multiple of 4, the derivatives of sin x corresponding to those numbers will be the same.

Answer

Since 12 is a multiple of 4, f ⁽¹²⁾(x) is the same as f ⁸(x) is the same as f ⁽⁴⁾(x) = sin x.

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22 differs from 2 by 20, which is a multiple of 4. Therefore

f ⁽²²⁾(x) = f ⁽²⁾(x) = -sin x.

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33 differs from 1 by 32, which is a multiple of 4, therefore

f ⁽³³⁾(x) = f ⁽¹⁾(x) = cos x.

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Some examples are e^x, sin x, cos x, and any constant (all the derivatives of a constant will be zero, but that still counts).