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Chapter 4: Problem 28

Find all functions \(f\) whose derivative is \(f^{\prime}(x)=x+1\).

Short Answer

Expert verified
Answer: \(f(x) = \frac{1}{2}x^2 + x + C\), where \(C\) is the constant of integration.

Step by step solution

01

Identify the given derivative

The problem states that the derivative of the function is \(f^{\prime}(x)=x+1\). We will integrate this with respect to \(x\) to find the original function \(f(x)\).
02

Integrate the derivative

To find the original function \(f(x)\), we need to integrate the given derivative \(f^{\prime}(x)\) with respect to \(x\). We can write this as: $$\int (x+1)dx$$
03

Perform the integration

Integrating the expression \((x+1)\) with respect to \(x\), we get: $$\int (x+1)dx = \frac{1}{2}x^2 + x + C$$ Where \(\frac{1}{2}x^2 + x\) is the antiderivative of \((x+1)\), and \(C\) is the constant of integration. It is important to include the constant of integration, as it represents all the possible vertical shifts of the function \(f(x)\).
04

Express the solution

Now we have found the function \(f(x)\), which is given by: $$f(x) = \frac{1}{2}x^2 + x + C$$ This function represents all possible solutions for the problem since the derivative of this function will always be \(f^{\prime}(x) = x+1\), regardless of the value of \(C\).

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Most popular questions from this chapter

Suppose you make a deposit of \(\$ P\) into a savings account that earns interest at a rate of \(100 \mathrm{r} \%\) per year. a. Show that if interest is compounded once per year, then the balance after \(t\) years is \(B(t)=P(1+r)^{t}\). b. If interest is compounded \(m\) times per year, then the balance after \(t\) years is \(B(t)=P(1+r / m)^{m t} .\) For example, \(m=12\) corresponds to monthly compounding, and the interest rate for each month is \(r / 12 .\) In the limit \(m \rightarrow \infty,\) the compounding is said to be continuous. Show that with continuous compounding, the balance after \(t\) years is \(B(t)=\overline{P e^{r t}}\).

Locate the critical points of the following functions and use the Second Derivative Test to determine whether they correspond to local maxima, local minima, or neither. $$f(x)=x^{3}+2 x^{2}+4 x-1$$

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Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=4 ; v(0)=-3, s(0)=2$$

Find the function \(F\) that satisfies the following differential equations and initial conditions. $$F^{\prime \prime \prime}(x)=672 x^{5}+24 x, F^{\prime \prime}(0)=0, F^{\prime}(0)=2, F(0)=1$$
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