Exam 6: Slope Fields and Eulers Method

The number of bacteria in a culture is increasing according to the law of exponential growth. After 5 hours there are 175 bacteria in the culture and after 10 hours there are 425 bacteria in the culture. Answer the following questions, rounding numerical answers to four decimal places. (i) Find the initial population. (ii) Write an exponential growth model for the bacteria population. Let $t$ represent time in hours. (iii) Use the model to determine the number of bacteria after 20 hours. (iv) After how many hours will the bacteria count be 15,000 ?

$\text { Find the exponential function } y = C e ^ { k x } \text { that passes through the two given points. }$ Round your values of $C$ and $k$ to four decimal places.

$\text { Determine whether the function } f ( x , y ) = x ^ { 4 } + 2 x ^ { 2 } y ^ { 2 } - 6 y ^ { 3 } \text { is homogeneous and }$ determine its degree if it is.

$\text { Sketch the slope field for the differential equation } \frac { d y } { d x } = 2 - y \text { and use the slope field }$ to sketch the solution satisfying the condition $y ( 0 ) = 6$ .

Write and solve the differential equation that models the following verbal statement. Evaluate the solution at the specified value of the independent variable, rounding your answer to four Decimal places: The rate of change of $P$ is proportional to $P$ . When $t = 0 , P = 60$ and when $t = 5 , P = 84$ . What is the value of $P$ when $t = 11$ ?