(10 points) Use the Fundamental Theorem of Calculus to find the exact values of the definite integral. \( \int_{1}^{3} 3\left(2^{x}\right) d x= \)

Determine the intervals on which the following function is concave up or concave down. Identify any inflection points. \( f(x)=x^{4}-3 x^{3}+9 \)

Find the intervals on which \( f \) is increasing and the intervals on which it is decreasing. \( f(x)=\frac{x^{4}}{4}-3 x^{3}+9 x^{2} \)

K For the following composite function, find an inner function \( u=g(x) \) and an outer function \( y=f(u) \) such that \( y=f(g(x)) \). Then calculate \( y=\tan 3 x^{5} \) Identify the inner and outer functions. Choose the correct answer. A. \( u=g(x)=x^{5} \) and \( y=f(u)=3 \tan u \) B. \( u=g(x)=\tan x \) and \( y=f(u)=3 u^{5} \) O. \( u=g(x)=\tan x^{5} \) and \( y=f(u)=3 u \) \( \frac{d y}{d x}=\square \)

Find an equation of the line tangent to the given curve at \( \mathrm{x}=\mathrm{a} \). \[ y=4+6 x+11 x e^{x} ; a=0 \] The equation for the tangent line is \( y=\square \).

Let \( f(x)=2 x^{3}+3 x^{2}-12 x+1 \). a. Find all points on the graph of \( f(x) \) at which the tangent line is horizontal. b. Find all points on the graph of \( f(x) \) at which the tangent line has slope 108 .