Multiple-Choice Questions
Please circle only one answer.

1. [2 marks] Let $f(x) = x^3 + \sqrt{x}$. Evaluate $f^{\prime}(1)$. In other words, find the derivative of $f$ at $x=1$.

   
   (a) $$f^{\prime}(1)=0$$ (b) $$f^{\prime}(1)=1.5$$ (c) $$f^{\prime}(1)=3.5$$ (d) $$f^{\prime}(1)=6$$

   
   

2. [2 marks] Let $$f(x) = \ln (\sqrt x)$$. Evaluate $f^{\prime\prime}(2)$. In other words, find the second derivative of $f$ at $x= 2$.

   
   (a) $$f^{\prime\prime}(2)= - \frac{1}{8}$$ (b) $$f^{\prime\prime}(2)=0$$ (c) $$f^{\prime\prime}(2)=-1$$ (d) $$f^{\prime\prime}(2)=\frac{3}{4}$$

   
   

3. [2 marks] Let $f(x) = \vert x-1\vert$. Calculate
   $$L = \lim_{h \to 0}\frac{f(1+h) - f(1)}{h}.$$
   (a) $L=0$ (b) This limit does not exist (c) $L=-1$ (d) $L=1$

   
   

4. [2 marks] Evaluate the following limit: $L = \displaystyle \lim_{x \to \infty} x \ \sin \left ( \frac{1}{x} \right)$.

   
   (a) $$L=1$$ (b) $$L=4$$ (c) $$L=2$$ (d) $$L=-1$$

   
   

5. [2 marks] A differentiable function $f$ has the property that $f(1) = 6$, $f^{\prime}(6)=-2$ and $f^{\prime}(1) = 3$. What is the value of the derivative of $f(f(x))$ at $x=1$?

   
   (a) $-3$ (b) $-6$ (c) $x$ (d) $2$

   
   

6. [2 marks] Let $$f(x) = \tan (1+ \sin x)$$. Evaluate $f^{\prime}(x)$. In other words, find the derivative of $f$ at $x$.

   
   (a) $$f^{\prime}(x)= -\sin x \ \sec^2 (\cos x)$$ (b) $$f^{\prime}(x)= \sec^2 (1+\sin x)$$ (c) $$f^{\prime}(x)= \sec^2 (\sin x)$$ (d) $$f^{\prime}(x)=\cos x \ \sec^2 (1+\sin x)$$

   
   

7. [2 marks] Let y be given implicitly as a differentiable function of $x$ by $x^2\cos y + y^2 - 1 = 0$. Then the slope of the tangent line to the curve $y = y(x)$ at the point $(x, y)$ where $x=0$, $y=1$ is equal to:

   
   (a) $$2$$, (b) $$+ \infty$$, (c) $$\frac{1}{2}$$, (d) $$0$$.

   
   

8. [2 marks] Let $$f(x) = \left(x+1\right )^{3x}$$. Evaluate $f^{\prime}(0)$. In other words, find the derivative of $f$ at $x=0$.

   
   (a) $$f^{\prime}(0)= 0$$ (b) $$f^{\prime}(0)=1$$ (c) $$f^{\prime}(0)= -1$$ (d) $$f^{\prime}(0)=12$$

   
   

9. [2 marks] Let $$f(x) = {\rm Arcsin} \left ( \cos x^2 \right)$$. Calculate $f^{\prime}(x)$ in the case where $x$ is a point where $\sin (x^2) > 0$.

   
   (a) $$f^{\prime}(x)= -5$$ (b) $$f^{\prime}(x)=x$$ (c) $$f^{\prime}(x)= -2x$$ (d) None of these

   
   

10. [2 marks] Evaluate $L = \displaystyle \lim_{x \to 0} \left ( \frac{\sin 4x}{\sin 2x} \right)$ using any method.

    
    (a) $L=1$ (b) $L = 1.85$ (c) $L =2$ (d) $$L = \frac{5}{2}$$

    
    

11. [2 marks] Evaluate
    $$\lim_{x \to +\infty}{d \over dx}\ \int_{\sqrt 3}^{\sqrt x}{\frac{r^3}{(r+1)(r-1)} \ dr}$$

    
    (a) $$I = 0$$
    (b) $$I = \frac{27}{2}$$
    (c) This limit does not exist (d) $$I = \frac{1}{2}$$

    
    

12. [2 marks] Which of the following functions $F$ represents the form of the inverse function of the function $f(x) = \sqrt{x^2-4}$ whose domain is the set of all real numbers $x$ where $x \geq 2$.

    
    (a) $$F(x) = \sqrt{x^2+4}$$ where $Dom(F) = \{ x\ : \ 0 \leq x < +\infty\ \}$
    (b) $$F(x) = - \sqrt{x^2+4}$$ where $Dom(F) = \{ x\ : \ -\infty < x < +\infty\ \}$
    (c) $$F(x) = \sqrt{4 + x^2}$$ where $Dom(F) = \{ x\ : \ -\infty < x < +\infty\ \}$
    (d) $$F(x) = \sqrt{x^2+ x +1}$$
    

    
    

13. [2 marks] Solve the inequality $$x^2-3x+2 < 0$$ for $x$.

    
    (a) $$\{ x : - 2 < x < -1 \}$$
    (b) $$\{ x : 1 < x < 2 \}$$
    (c) $$\{ x : -\infty < x < 1 \}$$
    (d) $$\{ x : 2 < x < \infty \}$$
    

    
    

14. Determine an interval where the graph of the function defined by the polynomial $p(x) = x^4 - 6x^3 +12x^2$ is concave up.

    
    (a) $$\{ x : - 5 < x < 0 \}$$
    (b) $$\{ x : -\infty < x < 1.78 \}$$
    (c) $$\{ x : 2 < x < \infty \}$$
    (d) $$\{ x : 1 < x < 2 \}$$
    

    
    

15. [2 marks] Which of the following functions has a point of inflection at $x=0$?

    
    (a) $$f(x) = x^2 + x + 1$$
    (b) $$f(x) = -x^2 - 4$$
    (c) $$f(x) = 2x^3 + 6$$
    (d) $$f(x) = x^4 + 12$$
    

    
    

16. [2 marks] For what values of $x$ is the function $$f(x) = \frac{1}{x^2-1}$$ increasing?

    
    (a) $$x > 0$$
    (b) $$x < 0$$
    (c) $$0 < x < 1$$
    (d) $$x > 1$$
    

    
    

17. [2 marks] Find all the critical points of $f(x) = x^3-3x+2$

    
    (a) $$x=-1, \ x=1$$
    (b) $$x=0, \ x=1$$
    (c) $$x=-2, x=0$$
    (d) $$x=0, \ x=1, \ x=2$$
    

    
    

18. [2 marks] An antiderivative of $f(x) = \cos (3x+6)$ is given by

    
    (a) $$\sin (3x+6)$$
    (b) $$-3 \sin (3x+6)$$
    (c) $$\frac{\sin (3x+6)}{3}$$
    (d) $$\sin (3x^2 + 6x)$$
    

    
    

19. [2 marks] Evaluate $$\int x^2\ 3^{2x^3+1} \ dx$$

    
    (a) $$\frac{\ln 3 \ 3^{2x^3+1}}{6}$$
    (b) $$6x^2 \ 3^{2x^3+1}$$
    (c) $$3^{2x^3+1}$$
    (d) $$\frac{3^{2x^3+1}}{6 \ln 3}$$
    

    
    

20. [2 marks] Evaluate $$\int_{0}^{4} x\ \sqrt{2x+1} \ dx$$

    
    (a) $$9$$
    (b) $$\frac{298}{15}$$
    (c) $$\frac{3}{4}$$
    (d) $$\frac{1}{2}$$
    

    
    

21. [2 marks] The value of $$\int {\frac{dx}{x\ln x}}$$ is

    
    (a) $$\ln ( \ln x ) \ +\ C$$
    (b) $$\ln \ln \vert x \vert \ +\ C$$
    (c) $$\ln \vert x\vert + C$$
    (d) $$\ln \vert \ln x \ \vert + C$$
    

    
    

22. [2 marks] The most general antiderivative of $x^2\ e^{3x}$ is given by

    
    (a) $$\frac{1}{3}x^2\ e^{3x}-\frac{2}{9}x\ e^{3x}+\frac{2}{27}\ e^{3x}\ +\ C$$
    (b) $$\frac{1}{9}x^3\ e^{3x} \ + \ C$$
    (c) $$\frac{1}{9}x^2\ e^{3x} + C$$
    (d) $$\frac{1}{3}x^2\ e^{3x}-\frac{2}{9}x\ e^{3x}+ C$$
    

    
    

23. [2 marks] Evaluate and simplify the indefinite integral: $$\int{e^{2x}\ \sin {3x}\ dx}$$.

    
    (a) $$- \frac{9}{13}\ e^{2x}\ \cos 3x - \frac{2}{13}\ e^{2x}\ \sin 3x\ +\ C$$
    (b) $$\frac{1}{13}\ e^{2x}\ \cos 3x + \frac{1}{13}\ e^{2x}\ \sin 3x\ +\ C$$
    (c) $$\frac{2}{13}\ e^{3x}\ \cos 2x + \frac{9}{13}\ e^{3x}\ \sin 2x\ +\ C$$
    (d) $$- \frac{3}{13}\ e^{2x}\ \cos 3x + \frac{2}{13}\ e^{2x}\ \sin 3x\ +\ C$$
    

    
    

24. [2 marks] Evaluate the improper integral $$\int_{0}^{\infty}{x^3\ e^{-x}\ dx}$$.

    
    (a) $$6$$
    (b) $$-8$$
    (c) $$0$$
    (d) $$1$$
    

    
    

25. [2 marks] The form of the partial fraction decomposition of the rational function
    $$\frac{x-1}{(x^2+1)(x^2 - 2x - 3)}$$
    is

    
    (a) $$\frac{A}{x^2+1} + \frac{C}{x-3}$$
    (b) $$\frac{Ax+B}{x^2+1} + \frac{C}{x-3} +\frac{D}{x+1}$$
    (c) $$\frac{A}{x-1} + \frac{B}{x+1} +\frac{C}{x-3}$$
    (d) $$\frac{Ax+B}{x^2+1} + \frac{C}{x+3} +\frac{D}{x+1}$$
    where $A, B, C, D$ are constants to be determined.

    
    

26. [2 marks] Evaluate the indefinite trigonometric integral
    $$\int{\sin^3 x \ \cos^2 x\ dx}.$$

    
    (a) $$\frac{\sin^4 x\ \cos^2 x}{4} +\ C$$
    (b) $$\frac{\sin^3 x\ \cos^3 x}{3} +\ C$$
    (c) $$\frac{\cos^5 x}{5} - \frac{\cos^3 x}{3} + \ C$$
    (d) $$\frac{\sin^5 x}{5} - \frac{\sin^3 x}{3} + \ C$$
    where $A, B, C, D$ are constants to be determined.

    
    

27. [2 marks] The area enclosed by the intersection of the two curves defined by $y = 1 - x$ and $y= 2x^2$ is given by which of the following definite integrals?

    
    (a) $$\int_{-1}^{\frac{1}{2}}{(2x^2 - 1) \ dx}$$
    (b) $$\int_{-1}^{\frac{1}{2}}{\left (1- x - 2x^2 \right) \ dx}$$
    (c) $$\int_{-1}^{\frac{1}{2}}{\left (2x^2 - x +1 \right) \ dx}$$
    (d) $$\int_{-1}^{2}{\left (x - 2x^2 \right) \ dx}$$
    

    
    

28. [2 marks] Which of the following expressions gives the volume of the solid of revolution obtained when the region bounded by the graphs of $y = 2x$ and $y = 4x^2$ is revolved about the $y-$axis?

    
    (a) $$I = \int_{0}^{\frac{1}{2}}{2\pi (2x - 4x^2)\ dx}$$
    (b) $$I = \int_{0}^{\frac{1}{2}}{2\pi x\ (4x^2-16x^4) \ dx}$$
    (c) $$I = \int_{0}^{\frac{1}{2}}{2\pi x\ (2x- 4x^2)\ dx}$$
    (d) $$I = \int_{0}^{1}{2\pi y \ (\sqrt y - y)\ dy}$$
    

    
    

29. [2 marks] The center of mass of a thin quarter circle of radius $R$ having uniform density and situated in the first quadrant is given by which of the following points?

    
    (a) $$\left (\frac{4\pi}{R}, \frac{4\pi}{R} \right )$$
    (b) $$\left (\frac{4R}{3\pi}, \frac{4R}{3\pi} \right )$$
    (c) $$\left (\frac{\pi}{R}, \frac{\pi}{R} \right )$$
    (d) $$\left (\frac{R}{\pi}, \frac{R}{\pi} \right )$$
    

    
    

30. [2 marks] The general solution of the differential equation
    $$y^2 \frac{dy}{dt} = t^2 e^{-y^3} \ln t$$
    is given by,
    
    

    
    (a) $$\frac{1}{3}e^{y^3} = \frac{1}{3}t^3 \ln t - \frac{t^3}{9} + C$$
    (b) $$e^{y^3} = \frac{1}{2}t^2 \ln {2t} +\frac{1}{9}e^{t} + C$$
    (c) $$\frac{1}{3}e^{y^3} = \frac{1}{2}t^2 \ln t - t + C$$ (d) $$e^{y^2}= \frac{1}{3}t^3 \ln t - \frac{t^3}{9} + C$$
    

    
    

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