1.Multiple Choice Questions: (only one correct answer for each of the following questions.)

(1) Suppose that

$${\int }_1^{5}f(x)dx=3,{\int }_5^{6}f(x)dx=2,{\int }_1^{6}g(x)dx=2$$

Then ${\int }_1^6(f(x)-2g(x))dx=$

(A) -1

(B) 1

(C) 3

(D) None of(A), (B)and(C)is correct

(2) How many real roots does the equation $x^3=2x^2+3x-3$ have?

(A) 0

(B) 1

(C) 2

(D) 3

(3) If $f(x)$ is twice differentiable on $[0,1]$ and $f^{\prime \prime }(x)>0$, then which of the following statements is correct?

(A) $f^{\prime }(1)>f^{\prime }(0)>f(1)-f(0)$

(B) $f^{\prime }(1)>f(1)-f(0)>f^{\prime }(0)$

(C) $f(1)-f(0)>f^{\prime }(1)>f^{\prime }(0)$

(D) $f^{\prime }(0)>f(1)-f(0)>f^{\prime }(1)$

(4) Let $f(x)=\{\begin{array}{ll}{x}^2\sin \frac{1}{x},& ,x\ne 0.\\ 0& ,x=0.\text{ Which of the following statements is correct? }\end{array}$

(A) $f^{\prime \prime }(0)=0$ and $(0,0)$ is a point of inflection

(B) $f^{\prime \prime }(0)=0$ and $(0,0)$ is not a point of inflection

(C) $f^{\prime \prime }(0)$ does not exist and $(0,0)$ is a point of inflection

(D) $f^{\prime \prime }(0)$ does not exist and $(0,0)$ is not a point of inflection

(5) Which of the following statements must be correct?

(A) If $f(x)$ is differentiable at $x=a$, then $|f(x)|$ is differentiable at $x=a$

(B) If $|f(x)|$ is differentiable at $x=a$, then $f(x)$ is differentiable at $x=a$

(C) If $f(x)$ is differentiable at $x=a$ and $f(a)=0,f^{\prime }(a)=0$, then $|f(x)|$ is differentiable at $x=a$

(D) If $f(x)$ is differentible at $x=a$ and $f(a)=0,f^{\prime }(a)\ne 0$, then $|f(x)|$ is differentiable at $x=a$

2.Fill in the blanks

(1) Let $f(x)=\{\begin{array}{ll}\sqrt{1-(x-1{)}^2},& \text{ if }0⩽x⩽2\\ -\sqrt{4-(x-4{)}^2},& \text{ if }2⩽x⩽6\end{array}$

According to the relationship between definite integral and area, ${\int }_0^{6}f(x)dx=$

(2) The linearization of $f(x)=\sqrt[3]{1+5x^4}$ at $x=0$ is $L(x)=$ $\underset{―}{}$

(3) Let $f(x)=x^3+3x+1$ .Use Newton＇s method to find the root of $f(x)=0$ .Start with $x_0=1$, then $x_2=$ $\underset{―}{}$

(4) If $f(x)+x\sin (f(x))=x^2+1$, then $f^{\prime }(0)=$ $\underset{―}{}$

(5) If $\underset{x\to \mathrm{\infty }}{lim}{f}^{\prime }(x)=10$, then $\underset{x\to \mathrm{\infty }}{lim}(f(x+10)-f(x))=$ $\underset{―}{}$

3.A rectangle is to be inscribed in the ellipse $\frac{x^2}{4}+y^2=1$

What should the dimensions of the rectangle be to maximize its area? What is the maximum area?

4. Let ${\frac{d}{dx}f(\sin x)|}_{x=0}={\frac{d}{dx}{f}^2(\sin x)|}_{x=0}$, and $f^{\prime }(0)\ne 0$. Find $f(0)$

5. Determine if the following limits exist or not. If so, find the limit. If not, explain why. (L'Hopital's Rule is not allowed to be used.)

(1) $\underset{x\to 1}{lim}\frac{x^4-1}{\sqrt{3+x}-2}$

(2) $\underset{x\to -\mathrm{\infty }}{lim}\frac{x^2\sin \frac{1}{2x}}{\sqrt{x^2+2024x+1}}$

6. Let $f(x)=x^{\frac{2}{3}}(6-x{)}^{\frac{1}{3}}$

(1) Indentify where the local extrema of $f$ occur. Find the function's local extreme values

(2) Find the open intervals where the graph of $f$ is concave up and where it is concave down

(3) Sketch the graph

7. Find $a\ne 0$ such that the curves $y=\frac{1}{2}{x}^2$ and $y^2+xy=a$ are tangent to each other, and find the equation of the tangent line at the point of tangency

8. Assume that $f$ is continuous on $[0,\mathrm{\infty })$, differentiable on $(0,\mathrm{\infty }),f(0)=0$ and $f^{\prime }$ is increasing on $(0,\mathrm{\infty })$. Prove that $\frac{f(x)}{x}$ is also increasing on $(0,\mathrm{\infty })$