What are the new limits of integration if apply the substitution \( u = 4 x + \pi \) to the integral \( \int _ { 0 } ^ { \pi } \sin ( 4 x + \pi ) d x \) ? (Express numbers in exact form. Use symbolic notation and fractions where needed.) lower limit: upper limit: 

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

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.)

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

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