Exam 3: Differentiation

Determine if the piecewise defined function is differentiable at the origin. - $f ( x ) = \left\{ \begin{array} { l l } 4 x - 6 & \text { if } x < 0 \\x ^ { 2 } + 6 x + 6 & \text { if } x \geq 0\end{array} \right.$

The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable? - $x = 1$

The graph of a function is given. Choose the answer that represents the graph of its derivative -

Find y'. - $y = \left( x + \frac { 1 } { x } \right) \left( x - \frac { 1 } { x } \right)$

Find the second derivative. - $r = \frac { 2 } { s ^ { 3 } } - \frac { 9 } { s }$

Graph the equation and its tangent - $\text { Graph } y = x ^ { 3 } - 4 \text { and the tangent to the curve at the point whose } x \text {-coordinate is } 0 \text {. }$

Provide an appropriate response. -Find all points $( x , y )$ on the graph of $f ( x ) = 2 x ^ { 2 } - 3 x$ with tangent lines parallel to the line $y = 13 x + 5$ .

Find the derivative of the function. - $y = \frac { x ^ { 2 } + 2 x - 2 } { x ^ { 2 } - 2 x + 2 }$

Estimate the slope of the curve at the indicated point. -

The graph of a function is given. Choose the answer that represents the graph of its derivative -

Find y'. - $y = \left( x ^ { 2 } - 3 x + 2 \right) \left( 5 x ^ { 3 } - x ^ { 2 } + 5 \right)$

Find the slope of the curve at the indicated point. -Find the points where the graph of the function have horizontal tangents. $f ( x ) = 5 x ^ { 2 } + 2 x - 4$

Provide an appropriate response. - $\text { Does the graph of } f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } \cos \frac { 1 } { x } , & x \neq 0 \\0 , & x = 0\end{array} \right. \text { have a tangent at the origin? Give reasons for your answer. }$

Given the graph of f, find any values of x at which f ' is not defined. -

Find the slope of the curve at the indicated point. -Find an equation of the tangent to the curve $f ( x ) = 2 x ^ { 2 } - 2 x + 1$ that has slope 2 .

Provide an appropriate response. -A charged particle of mass $m$ and charge $q$ moving in an electric field $E$ has an acceleration a given by $\mathrm { a } = \frac { \mathrm { qE } } { \mathrm { m } }$ , where $\mathrm { q }$ and $\mathrm { E }$ are constants. Find $\frac { \mathrm { da } } { \mathrm { dm } }$ .

Provide an appropriate response. -Find an equation for the tangent to the curve $y = \frac { 27 } { x ^ { 2 } + 2 }$ at the point $( 1,9 )$ .

Find an equation for the tangent to the curve at the given point. - $y = \frac { x ^ { 3 } } { 2 } , ( 8,256 )$

Provide an appropriate response. -Does the curve y = (x + 3)3 have any horizontal tangents? If so, where? Give reasons for your answer.

Find the second derivative. - $y = 6 x ^ { 4 } - 7 x ^ { 2 } + 2$