1.
Question 1. Use the rectangular approximation with 4 rectangles, using the left endpoint for the height, to approximate the area under the curve of \(f(x)=2x^2\) on the interval \([0,4]\).
Step 1: Divide \([0,4]\) into 4 equal parts so \(\Delta x=1\). Step 2: Using left endpoints gives \(x=0,1,2,3\). Step 3: Evaluate \(2x^2\): 0,2,8,18. Step 4: Multiply each by \(\Delta x=1\) and add: \(0+2+8+18=28\). Answer: 28
2.
Question 2. The sum \(\sum_{k=1}^{4}(5k+1)\) is equal to 54.
Step 1: Expand the sum: \((5\cdot1+1)+(5\cdot2+1)+(5\cdot3+1)+(5\cdot4+1)\). Step 2: Compute: \(6+11+16+21=54\). Step 3: Therefore the statement is true. Answer: True
3.
Question 3. If \(\int_5^8f(x)dx=5\) and \(\int_5^8g(x)dx=-2\), then \(\int_5^8(4f(x)+g(x))dx=22\).
Step 1: Apply linearity: \(4\int f+\int g\). Step 2: Substitute values: \(4(5)+(-2)=18\). Step 3: Since 18≠22, the statement is false. Answer: False
4.
Question 4. The integral \(\int_a^b3\sin x\,dx\) can be written as \(\lim_{\max\Delta x_k\to0}\sum3\sin(x_k^*)\Delta x_k\).
Step 1: Recall the definition of a definite integral as a limit of Riemann sums. Step 2: Here \(f(x)=3\sin x\). Step 3: Replace \(f(x_k^*)\) with \(3\sin(x_k^*)\). Step 4: The expression matches the definition. Answer: True
5.
Question 5. Find \(\frac{d}{dx}\int_3^x\frac{5t}{\ln(t+1)}dt\).
Step 1: Apply the Fundamental Theorem of Calculus. Step 2: Replace \(t\) with \(x\). Step 3: The derivative is \(\frac{5x}{\ln(x+1)}\). Answer: \(\frac{5x}{\ln(x+1)}\)
6.
Question 6. The area under the curve \(f(x)=x^{-1/3}\) on \([1,3]\) is 1.74.
Step 1: Since the function is positive, area equals the definite integral. Step 2: \(\int x^{-1/3}dx=\frac{3}{2}x^{2/3}\). Step 3: Evaluate: \(\frac32(3^{2/3}-1)\approx1.62\). Step 4: 1.62 is not 1.74. Answer: False
7.
Question 7. \(\int_1^4x\,dx=7.5\).
Step 1: Antiderivative of \(x\) is \(\frac{x^2}{2}\). Step 2: Evaluate from 1 to 4: \(\frac{16}{2}-\frac12=8-0.5=7.5\). Step 3: Therefore the statement is true. Answer: True
1 out of 1
