5.15 Slope fieldsIB Maths: Applications and Interpretation HL: Flashcards

What these 13 flashcards ask

What is a slope field?
How do you find the gradient of the segment at a point?
Gradient of the segment at (3,1) for \frac{dy}{dx}=x-y?
Where are segments horizontal?
For \frac{dy}{dx}=x-y, where are segments horizontal?
What is an isocline?
What is a solution curve?
Can two solution curves cross?
What does an initial condition do?
How do you find the tangent to the solution curve through (x0,y0)?
For \frac{dP}{dt}=0.2P(5-P), what do horizontal segments at P=0 and P=5 mean?
How do you describe the long-term behaviour from a slope field?
How do you check that a function is a solution?

Exam questions on 5.15 Slope fields

A slope field is drawn for the differential equation $\frac{dy}{dx}=x-y$ .
A solution curve passes through the point $(0,1)$ . Find the equation of the tangent to the solution curve at this point.2 marks
A slope field is drawn for the differential equation $\frac{dy}{dx}=\frac{x}{y}$ , for $y\neq0$ .
A solution curve passes through the point $(1,2)$ . Use the gradient of the slope field at this point to estimate the value of $y$ on this curve when $x=1.2$ .2 marks
The population $P$ , in hundreds, of fish in a lake at time $t$ years is modelled by $\frac{dP}{dt}=0.2P(5-P)$ for $P\ge0$ . A slope field is drawn for this differential equation.
Find the values of $P$ at which the line segments are horizontal, and interpret your answer in context.3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).