Question

Determine whether the following statements are true and give an explanation or counterexample. a. \(F(x)=x^{3}-4 x+100\) and \(G(x)=x^{3}-4 x-100\) are antiderivatives of the same function. b. If \(F^{\prime}(x)=f(x),\) then \(f\) is an antiderivative of \(F\) c. If \(F^{\prime}(x)=f(x),\) then \(\int f(x) d x=F(x)+C\) d. \(f(x)=x^{3}+3\) and \(g(x)=x^{3}-4\) are derivatives of the same function. e. If \(F^{\prime}(x)=G^{\prime}(x),\) then \(F(x)=G(x)\)

Short answer

Question: Determine whether each statement is true or false, and provide an explanation or counterexample. a. Statement: F(x)=x^3-4x+100 and G(x)=x^3-4x-100 are antiderivatives of the same function. Answer: True. Their derivatives are equal, so they are antiderivatives of the same function. b. Statement: If F'(x)=f(x), then f is an antiderivative of F. Answer: False. The correct statement is that F(x) is an antiderivative of f(x). c. Statement: If F'(x)=f(x), then ∫f(x)dx=F(x)+C. Answer: True. By definition, F(x) is an antiderivative of f(x), and the general antiderivative is given by F(x)+C. d. Statement: f(x)=x^3+3 and g(x)=x^3-4 are derivatives of the same function. Answer: False. Their antiderivatives differ by more than just a constant term, so they are not derivatives of the same function. e. Statement: If F'(x)=G'(x), then F(x)=G(x). Answer: False. The correct statement is that if F'(x)=G'(x), then F(x)=G(x) + C, where C is a constant.

Step-by-step solution

a. Statement: \(F(x)=x^{3}-4 x+100\) and \(G(x)=x^{3}-4 x-100\) are antiderivatives of the same function.

To check if they are antiderivatives of the same function, we need to calculate their derivatives and compare them. Let's take their derivatives using the power rule: \(\frac{dF(x)}{dx}=3x^2 - 4\) \(\frac{dG(x)}{dx}=3x^2 - 4\) Since both \(F'(x)\) and \(G'(x)\) are equal, the statement is true. The functions \(F(x)\) and \(G(x)\) are indeed antiderivatives of the same function.

b. Statement: If \(F^{\prime}(x)=f(x),\) then \(f\) is an antiderivative of \(F\)

This statement is false because the derivative and antiderivative are reverse operations. If \(F'(x)=f(x),\) then \(F(x)\) is an antiderivative of \(f(x),\) not the other way around.

c. Statement: If \(F^{\prime}(x)=f(x),\) then \(\int f(x) d x=F(x)+C\)

This statement is true by definition. If \(F'(x)=f(x),\) that means \(F(x)\) is an antiderivative of \(f(x),\) and the most general antiderivative of \(f(x)\) is given by \(F(x)+C,\) where \(C\) is the integration constant.

d. Statement: \(f(x)=x^{3}+3\) and \(g(x)=x^{3}-4\) are derivatives of the same function

In order to check whether these functions are derivatives of the same function, we need to find their antiderivatives and verify if they differ by a constant term. Let's integrate both functions: \(\int f(x) dx = \int (x^3 + 3) dx = \frac{1}{4} x^4 + 3x + C_1\) \(\int g(x) dx = \int (x^3 - 4) dx = \frac{1}{4} x^4 - 4x + C_2\) Now, we can compare the antiderivatives: \(F(x)=\frac{1}{4} x^4 + 3x + C_1\) \(G(x)=\frac{1}{4} x^4 - 4x + C_2\) Since the antiderivatives differ by more than just a constant term (\(3x\) and \(-4x\)), the statement is false. The functions \(f(x)\) and \(g(x)\) are not derivatives of the same function.

e. Statement: If \(F^{\prime}(x)=G^{\prime}(x),\) then \(F(x)=G(x)\)

This statement is false as it neglects the integration constant. If their derivatives are the same, that means they are antiderivatives of the same function, and thus they can only differ by a constant term. The correct statement would be: If \(F^{\prime}(x)=G^{\prime}(x),\) then \(F(x)=G(x) + C,\) where C is a constant.