Exam 5: Applications of Derivatives

Solve the problem. -A long strip of sheet metal 12 inches wide is to be made into a small trough by turning up two sides at right angles to the base. If the trough is to have maximum capacity, how many inches should be turned up on each Side?

Find the most general antiderivative. - $\int \left( \frac { 6 } { \sqrt { 1 - x ^ { 2 } } } - \frac { 7 } { x } \right) d x$

Solve the problem. -Use Newton's method to estimate the solution of the equation $3 \sin x - 4 x + 1 = 0$ . Start with $x 1 = 1.5$ . Then, in each case find $x _ { 2 }$ .

Estimate the limit by graphing the function for an appropriate domain. Confirm your estimate by using L'Hopital's rule. Show each step of your calculation. -Which one is correct, and which one is wrong? Give reasons for your answers. (a) $\lim _ { x \rightarrow 4 } \frac { x - 4 } { x ^ { 2 } - 4 } = \lim _ { x - 4 } \frac { 1 } { 2 x } = \frac { 1 } { 8 }$ (b) $\lim _ { x \rightarrow 4 x ^ { 2 } - 4 } \frac { x - 4 } { 12 } = \frac { 0 } { 12 } = 0$

Graph the function, then find the extreme values of the function on the interval and indicate where they occur. - $y = | x - 1 | - | x - 4 |$ on the interval $- 2 < x < 7$

Plot the zeros of the given polynomial on the number line together with the zeros of the first derivative. - $y = ( x - 5 ) ( x + 1 ) ^ { 2 }$

Find the limit. - $\lim _ { x \rightarrow 0 ^ { + } } x ^ { - 4 / \ln x }$

Graph the rational function. - $y = \frac { x - 2 } { x ^ { 2 } - 3 x + 2 }$

Find an antiderivative of the given function. - $5 \sqrt { x } - 7$

Use l'Hopital's Rule to evaluate the limit. - $\lim _ { x \rightarrow 1 } \frac { x ^ { 3 } - 9 x ^ { 2 } + 8 } { x - 1 }$

Solve the initial value problem. - $\frac { d y } { d x } = 3 x ^ { - 3 / 4 } , y ( 1 ) = 8$

Use differentiation to determine whether the integral formula is correct. - $\int ( 5 x + 6 ) ^ { - 2 } d x = - \frac { ( 5 x + 6 ) ^ { - 1 } } { 5 } + C$

Find an antiderivative of the given function. - $- \frac { 30 } { x ^ { 7 } }$

Solve the problem. -Find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph whe function has a local maximum or local minimum value. Then graph the function in a region large enough to shor these points simultaneously. Add to your picture the graphs of the function's first and second derivatives. $y = x ^ { 3 } - 15 x ^ { 2 }$

Find the largest open interval where the function is changing as requested. -Decreasing $\quad f ( x ) = | x - 8 |$

Find the extreme values of the function and where they occur. - $y = ( x + 1 ) ^ { 2 / 3 }$

Using the derivative of f(x) given below, determine the critical points of f(x). -f(x) = (x + 9)(x + 8)

Answer the problem. -Use the following function and a graphing calculator to answer the questions. $f ( x ) = x ^ { 4 } - 5 x ^ { 2 } + 4 x + 3 , [ - 0.5,1.8 ]$ a). Plot the function over the interval to see its general behavior there. Sketch the graph below. b). Find the interior points where f = 0 (you may need to use the numerical equation solver to approximate a solution). You may wish to plot f as well. List the points as ordered pairs (x, y). c). Find the interior points where f does not exist. List the points as ordered pairs (x, y). d). Evaluate the function at the endpoints and list these points as ordered pairs (x, y). e). Find the function's absolute extreme values on the interval and identify where they occur.

Determine all critical points for the function. - $f ( x ) = x ^ { 3 } - 12 x + 1$

Use l'Hopital's Rule to evaluate the limit. - $\lim _ { x - 0} \frac { \sin 5 x } { \sin x }$