An elevator with two electricians fixed on the floor The horizontal distance between them is m . The first elevator started to rise upwards at a speed of \( 3 \mathrm{~m} / \mathrm{s} \). After a second, the second elevator started to fall downwards at a speed of \( 2 \mathrm{~m} / \mathrm{s} \). Find the rate of change in the distance between them two seconds after the second elevator started to move?

Suppose \( \int_{2}^{4} f(x) d x=6, \int_{2}^{7} f(x) d x=-2 \), and \( \int_{2}^{7} g(x) d x=4 \). Evaluate the following integrals. \( \int_{7}^{2} g(x) d x=-4 \) (Simplify your answer.) \( \int_{7}^{7} 5 g(x) d x=20 \) 2 (Simplify your answer.) \( \int_{7}^{7}[g(x)-f(x)] d x=\square \) 2 (Simplify your answer.)

5. a) The function \( f(x)=x \sin (1 / x) \) is continuous for all \( x \neq 0 \). Show that if we define \( f(0)=0 \), then the function is continuous for all reals Hint: Squeeze Theorem b) Let \( g(x)=x^{2} \sin (1 / x) \) for \( x \neq 0 \) and \( g(0)=0 \). Show that \( g(x) \) is differentiable for all \( x \) but that \( g^{\prime}(x) \) is not continuous. Hint: sequences c) Suppose we have \( g(x) \) as defined in \( b) \) but \( g(0) \) was not given. Instead, we are given that \( g(x) \) is differentiable for all \( x \). Show that \( g^{\prime}(x)=0 \) for some point in the interval \( [-1 / \pi, 1 / \pi] \) Hint; Rolle's theorem d) Let \( H(x)=\int|t| \) dt where the limits are -17 to \( x \). Show that \( H(x) \) is continuously differentiable for all reals, but \( H^{\prime \prime}(x) \) does not exist at \( x=0 \). Hint: sequences

\( \int_{2}^{7} 5 g(x) d x=\square \) (Simplify your answer.)

What is the average rate of change on the function \( f(x)=x^{2} \) between \( x= \) 1 and \( x=3 \) ? -1

The president of a company that manufactures toy cars thinks that the function \( P(c)=-2 c^{2}+14 c-20 \) represents the company's profit, where \( c \) is the number of cars produced, in thousands, and \( P(c) \) is the company's profit, in hundreds of thousands of dollars. Determine the maximum profit the company can earn.