1.
Question 1. Find the local linear approximation of \(f(x)=e^{3x}\) at \(x=1\).
Step 1. Use \(L(x)=f(a)+f'(a)(x-a)\). Step 2. \(f(1)=e^3\), \(f'(x)=3e^{3x}\), so \(f'(1)=3e^3\). Step 3. Compute \(L(x)=e^3+3e^3(x-1)=3e^3x-2e^3\). Answer: C
2.
Question 2. Find the local linear approximation of \(f(x)=1/x\) at \(x=1\).
Step 1. Use linearization. Step 2. \(f(1)=1\), \(f'(1)=-1\). Step 3. \(L(x)=1-(x-1)=-x+2\). Answer: B
3.
Question 3. Approximate \(\int_0^3\sin x\,dx\) using the Trapezoidal Rule with \(n=4\).
Step 1. \(\Delta x=3/4\). Step 2. Apply the trapezoidal rule. Step 3. Evaluate to obtain approximately 1.896. Answer: B
4.
Question 4. Approximate \(\int_1^3 2^x\,dx\) using Simpson's Rule with \(n=4\).
Step 1. \(\Delta x=(3-1)/4=1/2\). Step 2. Apply Simpson's Rule. Step 3. Evaluate to obtain approximately 8.656. Answer: C
1 out of 1
