School of Mathematics and Statistics
Carleton University
Math. 69.104
TEST 1 SOLUTIONS

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PART I: Multiple Choice Questions
(Choose and CIRCLE only ONE answer - No part marks here.)

1. [3 marks] Let $f(x)=x^3 + \sqrt{{x^2+1}}$. Then $f^{\prime}(x)$ is equal to:

   
   $$3x^2 + \frac{x}{\sqrt{x^2+1}}$$, (b) $$3x^2 + \frac{x}{2\sqrt{x^2+1}}$$, (c) $$3x^2 + \frac{2x}{{\sqrt{x^2+1}}}$$, (d) $$3x^2$$.

   
   

2. [3 marks] Let $f(x)=x \cos x$. Which of the following expressions represents the value of $f^{\prime}(\pi)$?

   
   (a) $$\lim_{h\rightarrow 0}{\frac{f(h)-f(0)}{h}}$$, $$\lim_{h\rightarrow 0}{\frac{f(\pi+h)-f(\pi)}{h}}$$, (c) $$\lim_{h\rightarrow 0}{\frac{f(h)-f(1)}{h}}$$, (d) $$\lim_{h\rightarrow 0}{\frac{f(h)-f(-h)}{h}}$$.

3. [3 marks] Evaluate the limit: $$\lim_{x\rightarrow \infty }{\ \ \frac{1}{x^2}\ \sin \left( \frac{1}{x^2}\right) }.$$

   
   $$0$$ , (b) $$1$$ , (c) The limit does not exist, (d) $$+ \infty$$.

   
   

4. [3 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 of the curve $y = y(x)$ at the point $(x, y)$ where $x=0$, $y=1$ is equal to:

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

   
   

5. [3 marks] Answer TRUE or FALSE: The function $f$ defined by $f(x) = \vert x\vert + 2x$ is differentiable at $x=0$.

   
   (a) TRUE, FALSE

   
   

PART II: Show all work here.
No additional pages will be accepted

1. [6+7 marks] Evaluate the required derivative of each of the following functions: a) $$f(x) = \cos ({\rm Arctan}\ (x^3))$$. Find $f^{\prime}(0)$. b) $$f(x) = (\sin x)^3$$. Find $f^{\prime\prime}(x).$ Solution: a) Let $\Box = {\rm Arctan}\ x^3$. Then
   

   $$D(\cos ({\rm Arctan}\ x^3)) = D(\cos (\Box))$$

   $$= - \sin (\Box)\ D(\Box)$$

   $$= - \sin ({\rm Arctan}\ x^3) {\rm D(Arctan \ x^3)}$$

   $$= - \sin ({\rm Arctan}\ x^3) \ \frac{3x^2}{1+x^6}$$

   $$= - 3x^2 \frac{\sin ({\rm Arctan}\ x^3)}{1+x^6}.$$

   
   
   b) Let $\Box = {\sin x}$ . Then
   

   $$D\ (\sin x)^3 = D({\Box}^3)$$

   $$= 3{\Box}^2\ D(\Box)$$

   $$= 3 (\sin x)^2 {\cos x}.$$

   
   
   Next, just use the Chain Rule to find the next derivative:
   

   $$D\ 3 (\sin x)^2 {\cos x} = 3\ D \left ((\sin x)^2 {\cos x}\right )$$

   $$= 3\left ( - (\sin x)^3 + 2 \sin x (\cos x)^2 \right)$$

   
   
   Next,
2. [6 + 6 marks] a) Let $f(x) = 1 - x^2$, and $g(x) = \sin x$. Evaluate the composition $f(g(0))$ using any method. b) 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 $x=1$?

Solution: a) Let $\Box = g(x)$. Then, for a given $x$,

$$f(g(x)) = f(\Box)$$

$$= 1 -{\Box}^2$$

$$= 1 - (\sin x)^2$$

$$= (\cos x)^2$$

$$= 1, {\rm at} \ x = 0.$$

b) Let $\Box = f(x)$. Then,

$$Df(f(x)) = Df(\Box)$$

$$= f^{\prime}(\Box) D(\Box)$$

$$= f^{\prime}(f(x)) f^{\prime}(x)$$

$$= f^{\prime}(f(1)) f^{\prime}(1) , {\rm at} x=1,$$

$$= f^{\prime}(6) f^{\prime}(1)$$

$$= (-2)(3)$$

$$= - 6.$$

Total: /40