Exam 3: Differentiation Rules

x f(x) (x) g(x) (x) -6 7 -8 -6 7 -4 1 -5 0 5 -2 3 -2 4 3 0 5 0 6 1 2 -5 1 6 -1 4 -3 3 4 -3 6 1 5 0 -5 -Find $\frac { d } { d x } ( f ( x ) \cdot g ( x ) ) \text { when } x = - 2$

The displacement of a particle is given by $x = 0.03 \sin \left( 20 \pi t + \frac { \pi } { 2 } \right) \text { meters }$ . Find all times t > 0 where (a) The displacement attains its maximum value.(b) The velocity attains its maximum value.(c) The acceleration attains its maximum value.

Below is a table containing information about the differentiable functions f and g. x -2 -1 0 1 2 f(x) 4 3 5 2 1 (x) 1 4 2 5 3 g(x) 5 2 1 3 4 (x) 3 5 4 1 2 Suppose also that $F ( x ) = f ( x ) + g ( x )$ , $G ( x ) = f ( x ) \cdot g ( x )$ , and $H ( x ) = \frac { g ( x ) } { f ( x ) }$ .(a) Find $F ^ { \prime } ( 0 )$ (b) Find $G ^ { \prime } ( - 2 )$ (c) Find $H ^ { \prime } ( 2 )$

If $f ( x ) = x ^ { 4 } - 3 x$ , find $f ^ { \prime \prime } ( 1 )$ .

If the total cost for producing x units of a particular product is given by C(x), then the average cost of production those x units is given by $A ( x ) = \frac { C ( x ) } { x }$ (a) If our cost function is $C ( x ) = 15,000 + 100 x + 2 r ^ { 3 / 2 }$ , what value of x will result in the minimum average cost? (b) What is the minimum average cost?

Simplify the expression $\cos \left( 2 \sin ^ { - 1 } x \right)$ .

The position function for a particle moving along the x-axis at time t > 0 (in seconds) has its x-coordinate given by $x ( t ) = t ^ { 4 } - 8 t ^ { 3 } + 6 t ^ { 2 } + 40 t - 20$ .(a) Find a formula for the velocity and acceleration of the particle at any time t.(b) What is the velocity of the particle when t = 3? What does this value indicate?

x f(x) (x) g(x) (x) -6 7 -8 -6 7 -4 1 -5 0 5 -2 3 -2 4 3 0 5 0 6 1 2 -5 1 6 -1 4 -3 3 4 -3 6 1 5 0 -5 -Find $\frac { d } { d x } \left( e ^ { x } \cdot g ( x ) \right) \text { when } x = 0$

Consider the curve given by $x = 2 \sin t + 5 , y = 4 - 5 \cos t , 0 \leq t \leq 2 \pi$ Find $\frac { d y } { d x }$ at the point corresponding to $t = \frac { 5 \pi } { 4 }$

If $f ( x ) = \sin ^ { 2 } x , \text { find } f ^ { \prime } \left( \frac { \pi } { 4 } \right)$

The mass of a rod varies in such a way that the mass of a piece x meters long, measured from the left end, is $x ^ { 2 }$ kilograms. Find the density in kg/m at the point 2 meters from the left end.

If $f ( x ) = e ^ { x } g ( x ) \text {, and if } g ( 0 ) = 3 \text { and } g ^ { \prime } ( 0 ) = - 2 \text {, find } f ^ { \prime } ( 0 ) \text {. }$

Find $\frac { d y } { d x }$ , (a) $y = \sqrt { x ^ { 2 } - 5 x }$ (b) $y = \frac { \sin \left( x e ^ { x } \right) } { x ^ { 3 } + x }$ (c) $y = x ^ { 2 } \cdot e ^ { \sqrt { x } }$ (d) $y = e ^ { \sqrt { x ^ { 2 } + 1 } }$

If $x y - 2 y ^ { 2 } = x ^ { 3 } y$ find the value of $\frac { d y } { d x }$ at the point ( $- 2,3$ ).

Find an equation of the tangent line to the graph of $y = x \ln x$ at the point (1, 0).

If $x e ^ { y } = y x ^ { 2 }$ , find an expression for $\frac { d y } { d x }$ .

Let y = $x ^ { 2 }$ , x = 2, and $\Delta$ x = 1. Find the value of the differential dy.

Find $\frac { d y } { d x }$ for the parametric curve given by $x = t ^ { 4 } - t ^ { 2 } + t , y = \sqrt [ 3 ] { t }$

At how many different values of x does the curve y = x $3$ - 2x have a tangent line parallel to the line $y = x$ ?

x f(x) (x) g(x) (x) -6 7 -8 -6 7 -4 1 -5 0 5 -2 3 -2 4 3 0 5 0 6 1 2 -5 1 6 -1 4 -3 3 4 -3 6 1 5 0 -5 -Find $\frac { d } { d x } \left( \frac { g ( x ) } { f ( x ) } \right) \text { when } x = - 2$