[카이스트 미적분기출] 2008 Midterm Exam of Calculus I [더플러스수학]

 

1. Let \(\displaystyle f(x) = 2x + cos x \).

(a) (8 points) Verify that \(\displaystyle  f\) has an inverse.

(b) (8 points) Find \(\displaystyle  (f ^{−1} )  ( \pi)\).

 

2. (15 points) Let \(\displaystyle f(x) = \sqrt{4x+1}\). Use the formal \(\displaystyle \epsilon-\delta\) definition of limit with given \(\displaystyle \epsilon =1\) to explain that \(\displaystyle f \) is continuous at \(\displaystyle x = 2\).

 

 

 

3. (10 points) Does there exist a differentiable function \(\displaystyle f \) that has the value \(\displaystyle 1 \) only at \(\displaystyle x = 0,~ 2\), and \(\displaystyle 3\), and \(\displaystyle  f ' (x) = 0 \) only at \(\displaystyle x = −1, ~ \frac{3}{ 4}\), and \(\displaystyle \frac{3}{ 2}\). Support your answer.

 

 

4. Calculate the indicated limit.

(a) (8 points) \(\displaystyle \lim\limits_{ x \rightarrow 0+} \frac{1}{ x^ 2} \int_{0}^{x^2} \ln(\cos t) dt\).

(b) (8 points) \(\displaystyle \lim\limits_{ x \rightarrow \infty} \left( x^3 +1\right)^{\frac{1}{\ln x}}\).

(c) (8 points) \(\displaystyle \lim\limits_{ x \rightarrow \infty} \left( 1- \frac{5}{x} \right) ^x \).

 

 

5. (15 points) Assume that \(\displaystyle f \) is a continuous function and that

\(\displaystyle \int_{0}^{x}tf(t)dt =\sin \left(\frac{\pi}{2} x^2 \right)-\ln(x^2 +1).\)

Prove that \(\displaystyle f \) has a critical point \(\displaystyle c\) in the interval \(\displaystyle (−1, ~1)\).

 

 

 

6. Let \(\displaystyle f(x) = - \frac{\pi}{4}+\cos^{-1} \left( \frac{x}{2}\right)\).

(a) (6 points) What are the domain and the range of \(\displaystyle f(x)\)?

(b) (6 points) Find \(\displaystyle \sin(2 \cdot f(1)) \).

(c) (10 points) Find the linearization of \(\displaystyle f(x)\) at \(\displaystyle x = 1\).

(d) (5 points) Use Newton’s method to estimate a solution of \(\displaystyle f(x) = 0\). (It exists). Use the starting value \(\displaystyle x_1 = 1\) and compute \(\displaystyle x_2\).

 

 

7. The derivative \(\displaystyle  f'\) of a continuous function \(\displaystyle  f \) satisfies the followings:

(a) \(\displaystyle  f' (x)\) is decreasing on \(\displaystyle x < −3\)  and on \(\displaystyle x > 0\),

(b) \(\displaystyle  f' (x)\) is increasing on \(\displaystyle−3 < x < 0\),

(c) \(\displaystyle  f' (x) \geq 0\) on \(\displaystyle x ≤ 2\), 

(d) \(\displaystyle  f' (x)\leq 0\) on \(\displaystyle x \geq 2\).

Answer the followings:

(a) (5 points) Find the interval(s) where f is increasing and also concave down. Ans:

(b) (5 points) Does f have point(s) of inflection? Support your answer. Ans:

(c) (5 points) Does f have a local maximum? Support your answer.

 

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