AP Calculus BC 7.5 Approximating Solutions Using Euler’s Method - Exam Style Questions - MCQs - New Syllabus
Question
Let \( y=f(x) \) be the solution to the differential equation \( \dfrac{dy}{dx}=3y-4x \) with initial condition \( f(1)=2 \). What is the approximation for \( f(2) \) if Euler’s method is used, starting at \( x=1 \) with step size \( h=0.5 \)?
A) \( 3 \)
B) \( 4 \)
C) \( 4.5 \)
D) \( 5.5 \)
B) \( 4 \)
C) \( 4.5 \)
D) \( 5.5 \)
▶️ Answer/Explanation
Detailed solution
Step \(0\): \( (x_0,y_0)=(1,2) \), slope \( y’ = 3y-4x = 6-4=2 \).Update to \( x_1=1.5 \): \( y_1 = y_0 + h\,y'(x_0,y_0) = 2 + 0.5\cdot 2 = 3 \).
Step \(1\): slope at \( (1.5,3) \) is \( 3\cdot 3 – 4\cdot 1.5 = 9-6=3 \).
Update to \( x_2=2 \): \( y_2 = y_1 + h\,y'(x_1,y_1) = 3 + 0.5\cdot 3 = 4.5 \).
✅ Correct: C) \( 4.5 \)
Question
Let \(y=f(x)\) be the solution to the differential equation \(\displaystyle \frac{dy}{dx}=2x\!\left(y-x\right)\) with initial condition \(f(1)=0\).
What is the approximation for \(f(0)\) if Euler’s method is used, starting at \(x=1\) with two steps of equal size?
What is the approximation for \(f(0)\) if Euler’s method is used, starting at \(x=1\) with two steps of equal size?
(A) \(-\dfrac{9}{2}\)
(B) \(-\dfrac{7}{4}\)
(C) \(\dfrac{3}{4}\)
(D) \(2\)
▶️ Answer/Explanation
Correct Answer: C
Step size: \(h=\dfrac{0-1}{2}=-\tfrac{1}{2}\).
Start \((x_0,y_0)=(1,0)\). Slope \(y'(1,0)=2(1)\,(0-1)=-2\). \(y_1=y_0+h\,y'(1,0)=0+(-\tfrac{1}{2})(-2)=1,\quad x_1=0.5\).
Next slope \(y'(0.5,1)=2(0.5)\,(1-0.5)=1\times 0.5=0.5\). \(y_2=y_1+h\,y'(0.5,1)=1+(-\tfrac{1}{2})(0.5)=0.75=\dfrac{3}{4}\).
Hence \(f(0)\approx \dfrac{3}{4}\).