6) Prove using induction that for every positive integer \( n \), \( 1 \cdot 1!+2 \cdot 2!+3 \cdot 3!+\ldots+n \cdot n!=(n+1)!-1(5 \) points \( ) \)

The fox population in a certain region has a continuous growth rate of 4 percent per year. It is estimated that the population in the year 2000 was 22000 . (a) Find a function that models the population \( t \) years after \( 2000(t=0 \) for 2000). Hint: Use an exponential function with base \( e \). Your answer is \( P(t)= \)

Which function transforms the translating it up 5 units? (1 point) \( g(x)=-2^{x+5} \) \( g(x)=2^{-x+5} \) \( g(x)=-2^{x}+5 \) \( g(x)=2^{-x}+5 \)

\( f(x)=3 e^{x+1} \), for \( x=-4 \) Your answer is \( f(-4)=\square \)

Sketch the graph of a polynomial function with roots at \( 2 \) (multiplicity 1), \( -3 \) (multiplicity 2), and \( 4 \) (multiplicity 3). Describe how each root affects the shape of the graph.

Which of the following is true about the graph of \( f(x)=\sqrt{x+1} \) ? It has no \( y \)-intercept. It has a minimum value at \( x=-1 \). It has a vertical asymptote at \( x=1 \). It has no x-intercept