Problem 21 Consider the following functions... [FREE SOLUTION]

Chapter 5: Problem 21

Consider the following functions $f$ and real numbers a (see figure). a. Find and graph the area function $A(x)=\int_{a}^{x} f(t) d t$ b. Verify that $A^{\prime}(x)=f(x)$ $$f(t)=3 t+1, a=2$$

Short Answer

Expert verified

Question: Find the Area function $A(x)$ for the function $f(t)=3t+1$ and a real number $a=2$. Verify that $A^{\prime}(x)=f(x)$ and describe its graph. Answer: The Area function $A(x)$ for the function $f(t)=3t+1$ and a real number $a=2$ is given by $A(x)=\frac{3}{2}x^2+x-\left(\frac{3}{2}(2)^2+(2)\right)$. Its derivative, $A^{\prime}(x)$, is equal to the function $f(x) = 3x+1$. The graph of the Area function $A(x)$ is a upward-opening parabola, where the x-axis represents the input value $x$, and the y-axis represents the output value $A(x)$.

Step by step solution

Finding the Area function $A(x)$

Since $A(x)=\int_{a}^{x} f(t) d t$, we need to first integrate $f(t)$: $$\int f(t) dt = \int (3t+1) dt = \frac{3}{2}t^2 + t + C$$ Now, we can plug in the limits of integration to find $A(x)$: $$A(x)=\frac{3}{2}x^2+x-\left(\frac{3}{2}a^2+a\right)$$ Given that $a=2$, we can substitute the value of $a$ in the above expression: $$A(x)=\frac{3}{2}x^2+x-\left(\frac{3}{2}(2)^2+(2)\right)$$

Finding the derivative of the Area function $A(x)$

To verify that $A^{\prime}(x)=f(x)$, we'll find the derivative of $A(x)$: $$A^{\prime}(x) = \frac{d}{dx} \left(\frac{3}{2}x^2 + x - \left(\frac{3}{2}(2)^2+(2)\right)\right) = 3x + 1$$

Verifying that $A^{\prime}(x)=f(x)$

Now that we have found $A^{\prime}(x)$, we can compare it to the function $f(x)$. We have $A^{\prime}(x) = 3x + 1$ and $f(x) = 3t+1$. Setting $t = x$, we have $f(x) = 3x+1$. Thus, we can see that: $$A^{\prime}(x) = f(x)$$ The given statement is verified.

Graphing the area function $A(x)$

To graph the area function, we will use the equation we obtained in Step 1: $$A(x)=\frac{3}{2}x^2+x-\left(\frac{3}{2}(2)^2+(2)\right)$$ Plot this equation on the coordinate plane, with the x-axis representing the input value $x$, and the y-axis representing the output value $A(x)$: [label x-axis as "x", label y-axis as "A(x)"] [plot A(x) equation] In conclusion, we found the area function $A(x)$, verified that its derivative is equal to the function $f(x)$, and graphed the area function.

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91影视

Consider the following functions \(f\) and real numbers a (see figure). a. Find and graph the area function \(A(x)=\int_{a}^{x} f(t) d t\) b. Verify that \(A^{\prime}(x)=f(x)\) $$f(t)=3 t+1, a=2$$

Short Answer

Step by step solution

Finding the Area function \(A(x)\)

Finding the derivative of the Area function \(A(x)\)

Verifying that \(A^{\prime}(x)=f(x)\)

Graphing the area function \(A(x)\)

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