Find all points on the curve \( \ln (x y)=x^{2} \) where the tangent lines are vertical. Select one: a. None of the tangent lines are vertical. b. \( (0,0) \) c. \( (0,1) \) d. \( (1,0) \)

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

Evaluate the integral. \( \int_{1}^{4}\left(x^{2}-1\right) d x=\square \) (Simplify your answer.)

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

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

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. (Simplify your answer.) 7 \( \int_{2} 5 g(x) d x=20 \) (Simplify your answer.) 7 \( \int[g(x)-f(x)] d x=6 \) 2 \( (S i m p l i f y \) 7 \( \int_{2}^{7}[5 g(x)-f(x)] d x=\square \) 2