Exam 3: Short-Cuts to Differentiation

If g(t) represents the position of a particle at time t seconds, then g'(t)represents the __________ of the particle at time t seconds.

Find the derivative of $f ( x ) = e ^ { - ( 5 - x ) ^ { 2 } }$ .

The table below gives values for functions f and g, and their derivatives. -1 0 1 2 3 f 3 3 1 0 1 g 1 2 2.5 3 4 -3 -2 -1.5 -1 1 2 3 2 2.5 3 Find $\frac { d } { d x }$ g(f(x))at x = -1.If is cannot be computed from the information given, enter "cannot find".

Find f '(x)if $f ( x ) = e ^ { 6 x } \cos 4 x$ .

According to the Mean Value Theorem, if $f ( x ) = x ^ { 2 }$ then there exists a number c, $6 < c < 8$ , such that f '(c)= a.What is a?

Find the derivative of $h ( x ) = \left( 4 x ^ { 3 } + e ^ { x } \right) ^ { 3 }$ .

A table of values for a function F near x = 3 and tables of values for a function G near x = 3 and near x = 7 are given below.Estimate $F ^ { \prime } ( 3 )$ using the right-hand estimate. x 2.9 3.0 3.1 F(x) 6.7 7.0 7.3 G(x) 5.2 5.0 4.8 6.9 7.0 7.1 G(x) 0.95 1.00 1.05

The slope of the tangent line to the curve $y \cos x + e ^ { y } = 4$ at $\left( \frac { \pi } { 2 } , \ln 4 \right)$ is

The table below gives values for functions f and g, and their derivatives. -1 0 1 2 3 f 3 3 1 0 1 g 1 2 2.5 3 4 -3 -2 -1.5 -1 1 2 3 2 2.5 3 Find $\frac { d } { d x } ( f ( x ) g ( x ) )$ at x = 1.Round to 2 decimal places.

If $( x + y ) ^ { 2 } = ( 2 x + 3 ) ^ { 3 }$ , use implicit differentiation to find $\frac { d y } { d x }$ .

An arch over a lake has the form $y = 220 - 35 \cosh ( x / 35 )$ where x is the number of feet from a point on one side of the lake.What is the highest point on the arch?

Find the slope of the curve $x ^ { 4 } - 2 x y ^ { 2 } + y ^ { 5 } = 13$ at the point (2, 1).

Let f(x)and g(x)be two functions.Values of f(x), f '(x), g(x), and g'(x)for x = 0, 1, and 2 are given in the table below.Use the information in the table to find $H ^ { \prime } ( 2 )$ if $H ( x ) =$ [f(x)]². f(x) (x) g(x) (x) 0 1 -1 2 5 1 -1 2 4 0 2 7 3 11 0.5

Find the derivative of $y = \tanh ( 7 x )$ .

The derivative of $\frac { \left( z ^ { 2 } + 1 \right) ^ { 2 } } { z }$ is $3 z ^ { 2 } + 2 - \frac { 1 } { z ^ { 2 } }$ .

Consider the equation ln x = mx where m is some constant (positive, negative, or zero).How many solutions will the equation have for m = 0.439?

If a and b are constants and $g ( x ) = \left( \alpha x ^ { 2 } + b \right) ^ { 2 }$ , find $g ^ { \text {ttt} } ( x )$ .

A man plans to propose to a woman in romantic fashion by taking her up in an air balloon.Unfortunately, he pulls the diamond ring from his pocket and drops it over the side of the balloon's basket.The ring's position above the earth t seconds after it falls is given by the function $s(t)=-16 t^{2}+1225$ feet.How fast is the ring falling 3 seconds after he drops it?

The derivative of $\frac { t } { e ^ { t } } + e ^ { 6 }$ is $\frac { 1 - t } { e ^ { t } } + e ^ { 6 }$ .

Find the equation of the line that is "orthogonal" to the graph of the function $f ( t ) = \sin ( t )$ at the point where $t = \frac { \pi } { 2 }$ .In other words, find the line perpendicular to the tangent line when $t = \frac { \pi } { 2 }$ .