Quiz Chapter 02.01: Primer on Differential Calculus

1. The definition of the first derivative of a function $f \left( x \right)$ is

{f}' \left( x \right) =\dfrac{f \left( x+\Delta x \right )+f \left(x \right )]}{\Delta x}

{f}' \left( x \right) =\dfrac{f \left( x+\Delta x \right )-f \left(x \right )]}{\Delta x}

{f}' \left( x \right) \lim_{x\rightarrow 0}\dfrac{f \left( x+\Delta x \right )+f \left(x \right )]}{\Delta x}

{f}' \left( x \right) \lim_{x\rightarrow 0}\dfrac{f \left( x+\Delta x \right ) - f \left(x \right )]}{\Delta x}

2. Given $y=5e^{3x}+\sin{\left( x \right)}$ , then $dy/dx$ is

5e^{3x}+\cos{\left( x \right)}

15e^{3x}+\cos{\left( x \right)}

5e^{3x}-\cos{\left( x \right)}

2.666e^{3x}-\cos{\left( x \right)}

3. Given $y=\sin{\left( 2x \right)}$ , then $dy/dx$ at $x=3$ most nearly is

0.9600

0.9945

1.920

1.989

4. Given $y=x^{3}\ln{\left( x \right)}$ , then $dy/dx$ is

3x^{2} \ln{\left( x \right)}

3x^{2} \ln{\left( x \right)} + x^{2}

x^{2}

3x

5. The velocity of a body as a function of time is given as $v \left( t \right) = 5e^{-2t} + 4$ , where $t$ is in seconds, $v$ is in $m/s$ . The acceleration at $t=0.6$ in $m/s^{2}$ is

-3.012

5.506

4.147

-10

6. If $x^{2} + 2xy = y^{2}$ , then $dy/dx$ is

\left( x+y \right) / \left( y-x \right)

2x + 2y

\left( x+1 \right) / y

-x