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AP Calculus BC Practice Quiz: Approximating Solutions Using Euler's Method (BC ONLY)

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 7 questions to check your progress.

Question 1 of 7

Let y = f(x) be the particular solution to the differential equation dy/dx = 2x with the initial condition f(1) = 3. What is the approximation for f(1.5) obtained by using Euler's method with a single step of size 0.5?

All Questions (7)

Let y = f(x) be the particular solution to the differential equation dy/dx = 2x with the initial condition f(1) = 3. What is the approximation for f(1.5) obtained by using Euler's method with a single step of size 0.5?

A) 3.5

B) 4.0

C) 4.5

D) 5.0

Correct Answer: B

Euler's method uses the formula y_1 ≈ y_0 + h * f'(x_0, y_0). Here, (x_0, y_0) = (1, 3), h = 0.5, and dy/dx = 2x. First, calculate the slope at the initial point: dy/dx at x=1 is 2(1) = 2. Then, apply the formula: y_1 ≈ 3 + 0.5 * (2) = 3 + 1 = 4.0.

Consider the differential equation dy/dx = y with the initial condition y(0) = 1. Use Euler's method with two steps of equal size to approximate y(0.2).

A) 1.20

B) 1.21

C) 1.22

D) 1.24

Correct Answer: B

The total interval is 0.2 and there are two steps, so the step size h = 0.2 / 2 = 0.1. Step 1: Start at (x_0, y_0) = (0, 1). The slope is dy/dx = y = 1. The new y-value is y_1 = y_0 + h * (slope) = 1 + 0.1 * (1) = 1.1. The new point is (0.1, 1.1). Step 2: Start at (x_1, y_1) = (0.1, 1.1). The slope is dy/dx = y = 1.1. The new y-value is y_2 = y_1 + h * (slope) = 1.1 + 0.1 * (1.1) = 1.1 + 0.11 = 1.21.

Euler's method is a procedure for approximating solutions to differential equations. The method works by iteratively stepping along the direction of the...

A) Secant line

B) Normal line

C) Tangent line

D) Horizontal asymptote

Correct Answer: C

Euler's method provides an approximation by using the tangent line at a known point on the curve to estimate the next point. The differential equation gives the slope of the tangent line at any point on the solution curve.

Let y = f(x) be the solution to the differential equation dy/dx = x + y with the initial condition f(0) = 1. What is the approximation for f(1) obtained by using Euler's method with two steps of equal size?

A) 2.00

B) 2.25

C) 2.50

D) 2.75

Correct Answer: C

With an interval from x=0 to x=1 and two steps, the step size h = (1-0)/2 = 0.5. Step 1: Start at (x_0, y_0) = (0, 1). The slope is dy/dx = x + y = 0 + 1 = 1. The new y-value is y_1 = y_0 + h * (slope) = 1 + 0.5 * (1) = 1.5. The new point is (0.5, 1.5). Step 2: Start at (x_1, y_1) = (0.5, 1.5). The slope is dy/dx = x + y = 0.5 + 1.5 = 2. The new y-value is y_2 = y_1 + h * (slope) = 1.5 + 0.5 * (2) = 1.5 + 1 = 2.50.

Let y(x) be the exact solution to a differential equation, and let y_E be the approximation found using Euler's method. If the graph of y(x) is known to be concave up on an interval, what is the relationship between the exact solution and the approximation on that interval?

A) The approximation y_E will be an underestimate of y(x).

B) The approximation y_E will be an overestimate of y(x).

C) The approximation y_E will be exactly equal to y(x).

D) The relationship cannot be determined without knowing the differential equation.

Correct Answer: A

Euler's method uses the tangent line at a point to approximate the function's value at the next step. For a function that is concave up, its graph lies above all of its tangent lines. Therefore, each step of Euler's method will fall below the actual solution curve, resulting in an underestimate.

The table below shows the first step of Euler's method used to approximate f(0.2), where y=f(x) is the solution to the differential equation dy/dx = x - 2y with initial condition f(0)=4. What is the value of y_1, the approximation for f(0.1)? | n | x_n | y_n | dy/dx at (x_n, y_n) | Δy = (h)(dy/dx) | |---|---|---|---|---| | 0 | 0.0 | 4.0 | -8.0 | -0.8 | | 1 | 0.1 | y_1 | | |

A) -8.0

B) -0.8

C) 3.2

D) 4.8

Correct Answer: C

The table shows the values for the first step (n=0) and asks for the result of that step, which is y_1. The formula for Euler's method is y_{n+1} = y_n + Δy. From the table, y_0 = 4.0 and the change in y, Δy, is calculated as -0.8. Therefore, y_1 = 4.0 + (-0.8) = 3.2.

Let y = f(x) be the solution to the differential equation dy/dx = 1/x with f(1) = 0. Using Euler's method with a step size of h = 0.5, the first approximation is f(1.5) ≈ 0.5. What is the value of the slope used to calculate the next approximation, f(2.0)?

A) 1/2

B) 2/3

C) 1

D) 2

Correct Answer: B

The procedure for the second step of Euler's method begins at the point found in the first step. The first step starts at (1, 0) and ends at the approximated point (1.5, 0.5). To find the next approximation for f(2.0), we must calculate the slope at this new point (1.5, 0.5). The slope is given by dy/dx = 1/x. Evaluating at x = 1.5 gives a slope of 1/1.5, which is equal to 2/3.