91Ó°ÊÓ

Find the marginal cost for producing units. (The cost is measured in dollars.) $$ C=205,000+9800 x $$

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=x^{3}+2 x ;(1,3) $$

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=\sqrt{x+1} ;(8,3) $$

Health The temperature \(T\) (in degrees Fahrenheit) of a person during an illness can be modeled by the equation \(T=-0.0375 t^{2}+0.3 t+100.4,\) where \(t\) is time in hours since the person started to show signs of a fever. (a) Use a graphing utility to graph the function. Be sure to choose an appropriate window. (b) Do the slopes of the tangent lines appear to be positive or negative? What does this tell you? (c) Evaluate the function for \(t=0,4,8,\) and \(12 .\) (d) Find \(d T / d t\) and explain its meaning in this situation. (e) Evaluate \(d T / d t\) for \(t=0,4,8,\) and \(12 .\)

Use the limit definition to find an equation of the tangent line to the graph of \(f\) at the given point. Then verify your results by using a graphing utility to graph the function and its tangent line at the point. $$ f(x)=\frac{1}{2} x^{2} ;(2,2) $$

Marginal Profit When the price of a glass of lemonade at a lemonade stand was \(\$ 1.75,400\) glasses were sold. When the price was lowered to \(\$ 1.50,500\) glasses were sold. Assume that the demand function is linear and that the variable and fixed costs are \(\$ 0.10\) and \(\$ 25\), respectively. (a) Find the profit \(P\) as a function of \(x,\) the number of glasses of lemonade sold. (b) Use a graphing utility to graph \(P,\) and comment about the slopes of \(P\) when \(x=300\) and when \(x=700\). (c) Find the marginal profits when 300 glasses of lemonade are sold and when 700 glasses of lemonade are sold.

A ball is propelled straight upward from ground level with an initial velocity of 144 feet per second. (a) Write the position, velocity, and acceleration functions of the ball. (b) When is the ball at its highest point? How high is this point? (c) How fast is the ball traveling when it hits the ground? How is this speed related to the initial velocity?

A brick becomes dislodged from the top of the Empire State Building (at a height of 1250 feet) and falls to the sidewalk below. (a) Write the position, velocity, and acceleration functions of the brick. (b) How long does it take the brick to hit the sidewalk? (c) How fast is the brick traveling when it hits the sidewalk?

Political Fundraiser A politician raises funds by selling tickets to a dinner for \(\$ 500 .\) The politician pays \(\$ 150\) for each dinner and has fixed costs of \(\$ 7000\) to rent a dining hall and wait staff. Write the profit \(P\) as a function of \(x,\) the number of dinners sold. Show that the derivative of the profit function is a constant and is equal to the increase in profit from each dinner sold.

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \text { If } f(x)=g(x)+c, \text { then } f^{\prime}(x)=g^{\prime}(x) $$