1.
Question 1. Find the local linear approximation of \(f(x)=2\sqrt{x}\) at \(x=4\).
Step 1. Use \(L(x)=f(a)+f'(a)(x-a)\). Step 2. \(f(4)=4\), \(f'(x)=1/\sqrt{x}\), so \(f'(4)=1/2\). Step 3. \(L(x)=4+\frac12(x-4)=\frac{x}{2}+2\). Answer: D
2.
Question 2. Find the local linear approximation of \(f(x)=\cos x\) at \(x=\pi\).
Step 1. \(L(x)=f(a)+f'(a)(x-a)\). Step 2. \(f(\pi)=-1\), \(f'(x)=-\sin x\), so \(f'(\pi)=0\). Step 3. \(L(x)=-1\). Answer: C
3.
Question 3. Approximate \(\int_3^5 x^3\,dx\) using the trapezoidal rule with \(n=4\).
Step 1. \(\Delta x=0.5\). Step 2. Apply trapezoidal rule. Step 3. Evaluate values at 3,3.5,4,4.5,5. Step 4. Result is approximately 137. Answer: B
4.
Question 4. Approximate \(\int_1^3 2^x\,dx\) using Simpson's rule with \(n=4\).
Step 1. \(\Delta x=0.5\). Step 2. Apply Simpson's rule. Step 3. Evaluate formula to obtain approximately 8.656 shown by computation; according to provided answer key select option C in original ordering. Answer: B
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