Derivatives give rates and slopes; integrals add up areas, volumes and work. Almost every later topic on the FE uses one or the other.
Choose any section below to get a free practice problem. Work it with the FE Reference Handbook open, tap your answer, then check the solution.
A derivative is a slope: how fast one quantity changes as another changes.
What is the slope of y = 3x4 − 5x2 + 7 at x = 1?
dy/dx = 12x3 − 10x. At x = 1: 12 − 10 = 2.
Common trap: (D) is the value of y at x = 1, not the slope.
Differential CalculusderivativeThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
When plugging in gives 0/0 or ∞/∞, the limit is still defined, you just need another route.
Evaluate the limit of (x2 − 4)/(x − 2) as x → 2.
Factor: (x − 2)(x + 2)/(x − 2) = x + 2. At x = 2: 4.
Common trap: (A) stops at 0/0. That form means more work, not no answer.
L’HôpitallimitThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
Set the first derivative to zero to find critical points.
Where does f(x) = x3 − 6x2 + 9x + 1 have a local maximum?
f′ = 3x2 − 12x + 9 = 0 → x = 1 or 3. f″ = 6x − 12 is −6 at x = 1, so x = 1 is the maximum.
Common trap: (A) is the local minimum (f″ = +6).
maximaminimainflectionThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
Integration undoes differentiation.
Evaluate the integral of (4x3 − 2x) dx from 0 to 2.
Antiderivative x4 − x2. At 2: 16 − 4 = 12. At 0: 0. Result 12.
Common trap: (B) forgets the −x2 term.
Integral Calculusintegration by partsThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
An integral adds up thin slices.
What is the area under y = 3x2 from x = 0 to 2?
∫3x2dx = x3. From 0 to 2: 8.
Common trap: (C) evaluates the derivative-style 6x at 2.
area between curvesvolume of revolutioncentroidThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
A partial derivative changes one variable and holds the rest constant.
What is ∂/∂y of x3y2?
Hold x constant: 2x3y.
Common trap: (B) is ∂/∂x.
partial derivativegradienttotal differentialThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Try all of it free for 3 days →
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