Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
These 20 problems cover AP Calculus AB Unit 7: verifying solutions, slope fields, separation of variables, exponential growth and decay, and particular solutions from initial conditions. Early problems verify or evaluate; later ones solve full models.
The last six problems are applied or multi-step, covering mixing, draining, cooling and the choice of sign when solving for a root.
Standards: AP Calculus AB Unit 7 · Level: Grade 12 AP Calculus AB
Problem 1 Verifying an exponential solution ★★★
Show that y = 3e2x is a solution of dydx = 2y.
Problem 2 General and particular solutions ★★★
Show that y = x2 + C solves dydx = 2x, and find the particular solution with y(1) = 5.
Problem 3 Verifying a second-order solution ★★★
Show that y = 5 sin 2x + x satisfies d2ydx2 + 4y = 4x.
Problem 4 Computing slopes from an equation ★★★
For dydx = x + y, find the slope at (1, 2), at (−2, 1) and at (0, 0).
Problem 5 Reading a slope field ★★★
The slope field for dydx = x − y is shown. Along which line are all the slopes zero? Is the solution through (0, 1) increasing or decreasing at x = 0?
Problem 6 Separable equation with a power ★★★
Find the particular solution of dydx = 4x3y with y(0) = 5.
Problem 7 Separable equation with a quotient ★★★
Solve dydx = xy with y(2) = 3.
Problem 8 Population growth model ★★★
A bacteria culture satisfies dP/dt = 0.05P with P(0) = 400, where t is in hours. Find P(t) and P(10).
Problem 9 Half-life ★★★
A 90-milligram sample of a medicine decays at a rate proportional to the amount present, with a half-life of 8 days. How many milligrams remain after 20 days?
Problem 10 Equation with an exponential ★★★
Find the particular solution of dydx = ex − y with y(0) = ln 3.
Problem 11 Equation with a trigonometric factor ★★★
Find the solution of dydx = y sin x that satisfies y(0) = 3.
Problem 12 Newton’s law of cooling ★★★
A cup of coffee at 190°F is placed in a 70°F room. Its temperature T obeys dT/dt = −0.1(T − 70) with t in minutes. Find T(t) and the temperature after 10 minutes.
Problem 13 Equilibrium and long-term behavior ★★★
The slope field for dydx = y(4 − y) is shown. Name the equilibrium solutions, and describe the limit of the solution with y(0) = 1 as x → ∞. Where is that solution concave up?
Problem 14 Separable equation with a cube root ★★★
Find the particular solution of dydx = (x + 1)2y2 with y(0) = 2.
Problem 15 Fish population ★★★
A lake has 2,400 fish, and the population grows at a rate proportional to its size. After 3 years there are 3,100. Find the growth constant k and predict the population after 10 years.
Problem 16 Salt in a tank ★★★
A 300-gallon tank starts with pure water. Salt solution enters, adding 6 pounds of salt per minute, and the well-mixed solution leaves so that the salt amount S (pounds) obeys dS/dt = 6 − S/50. Find S(t), the amount after 50 minutes and the long-term amount.
Problem 17 Draining tank ★★★
Water drains from a tank so that its depth h (feet) satisfies dh/dt = −0.3√h with h(0) = 16, t in minutes. Find h(t) and the time when the tank is empty.
Problem 18 Choosing the sign of a root ★★★
Solve dydx = x + 2y with y(0) = −1.
Problem 19 Equilibrium and exponential behavior ★★★
Solve dydx = x(y − 3) with y(0) = 5. Identify the equilibrium solution and find the value of y(2).
Problem 20 Time to cool ★★★
Tea at 96°F sits in a 22°F porch. Its temperature satisfies dT/dt = −0.08(T − 22). When does the tea reach 50°F? Will it ever reach 22°F?
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