To find the derivative of a function \( f(x) \), graphing calculators use the formula \( f^{\prime}(x)=\lim \delta \rightarrow 0[f(x+\delta)-f(x-\delta)] / 2 \delta \). For the second derivative, the formula is \( f^{\prime \prime}(x)=\lim \delta \rightarrow 0[f(x+\delta)+f(x-\delta)-2 f(x)] / \delta^{2} \) a) Use the first formula to find the derivative of \( f(x)=x^{3} \) and \( g(x)=|x| \) b) Use the second formula to find the second derivative of \( h(x)=x^{4} \) and \( s(x)=\sin (1 / x) \) c) If \( f(x+\delta)+f(x-\delta)-2 f(x)>0 \) for all \( x \), is \( f(x) \) concave up or concave down?

What is the average distance between the parabola \( y=7 x(18-x) \) and the \( x \)-axis on the interval \( [0,18] \) ? The average distance is (Type an integer, proper fraction, or mixed number)

The function \( h(t)=1+4 t-1.86 t^{2} \) models the height of a rock thrown upward on the planet Mars, where \( h(t) \) is height in metres and \( t \) is time in seconds. Use a graph to determine a) the maximum height the rock reaches b) how long the rock will be above the surface of Mars

1. The Laplace transoform of a causal signal \( x(t) \) is \[ X(s)=\frac{s+5}{s^{2}+3 s+2}, \operatorname{Re}(s)>-1 \] Using the properties of Laplace transform, find the Laplace transform of the following signals (a) \( 3 x(t / 3) \) (b) \( x(t-2) \) (c) \( (t-1) x(t) \) (d) \( \frac{d x(t)}{d t} \) (e) \( x(t) \exp (-2 t) \) (f) \( x(t) \cos (2 t) \)

\( y ^ { \prime \prime } + y = x ^ { 3 } + 1 \)

The elevation of a path is given by \( f(x)=x^{3}-6 x^{2}+4 \), where \( x \) measures horizontal distances. Draw a graph of the elevation function and find its average value for \( 0 \leq x \leq 4 \).