▶️ Answer/Explanation
Solution
Correct Answer: B
1. As x→∞, the numerator grows as -6x² (polynomial growth)
2. The denominator grows as 3eˣ (exponential growth)
3. Exponential functions dominate polynomial functions as x→∞
4. Therefore, the limit approaches 0
5. The limit exists and equals 0 (Option B)
▶️ Answer/Explanation
Solution
Correct Answer: C
1. For option C: \(y=\frac{x}{x+1}\)
2. Horizontal asymptote found by \(\lim_{x\to\pm\infty}\frac{x}{x+1} = 1\)
3. Other options:
– A) No horizontal asymptote
– B) Oscillates between -1 and 1
– D) Has asymptote at x=1 but y approaches ±∞
– E) Approaches 0 as x→∞
The graph of a function f is shown above. If g is the function defined by: \[ g(x)=\frac{x^{2}+1}{f(x)} \] what is the value of g'(2)?▶️ Answer/Explanation
Solution
Correct Answer: A
1. Apply the quotient rule: \(g'(x)=\frac{2xf(x)-f'(x)(x^{2}+1)}{(f(x))^{2}}\)
2. From the graph:
\(f(2) = 3\)
\(f'(2) = \frac{7-3}{3-2} = 4\) (slope between points (2,3) and (3,7))
3. Evaluate at x=2:
\[g'(2)=\frac{4(3)-4(5)}{9}=\frac{12-20}{9}=-\frac{8}{9}\]
▶️ Answer/Explanation
Solution
Correct Answer: A
1. We know: \(\int_{13}^{20} f(x) \, dx = 7\)
2. Break this into two parts: \(\int_{13}^{17} f(x) \, dx + \int_{17}^{20} f(x) \, dx = 7\)
3. Substitute known value: \(\int_{13}^{17} f(x) \, dx + (-3) = 7\)
4. Solve: \(\int_{13}^{17} f(x) \, dx = 10\)
5. Now consider \(\int_0^{17} f(x) \, dx = \int_0^{13} f(x) \, dx + \int_{13}^{17} f(x) \, dx\)
6. Substitute known values: \(8 = \int_0^{13} f(x) \, dx + 10\)
7. Final calculation: \(\int_0^{13} f(x) \, dx = 8 – 10 = -2\)
