SME
College · Math
Calculus I
8 topics
Before this course
Limits, continuity and an introduction to derivatives
Limits of sequences and the area under a curve
The unit circle and radian measure
Exponential and logarithmic equations, e and continuous growth
Topics
Limits and continuity
Main lesson
Make a piecewise function continuous
Use this for “Find k so that f(x) = 3x + k (x < 2), x² + 1 (x ≥ 2) is continuous.”
The intermediate value theorem
Use this for “Show that x³ + x − 1 = 0 has a root between 0 and 1.”
The limit of sin(kx) ÷ (mx)
Use this for “Find the limit of sin(3x) ÷ (2x) as x → 0” or “Find k so that sin(kx) ÷ (2x) → 3 as x → 0.”
Find δ for a given ε (a line)
Use this for “For lim (3x − 1) = 5 as x → 2, find δ when ε = 0.06.”
Limits at infinity of a quotient
Use this for “Find the limit of (5x² − x) ÷ (2x² + 7) as x → ∞.”
Kinds of discontinuity
Use this for “Is the discontinuity of (x² − 4) ÷ (x − 2) at x = 2 removable?”
Derivatives and differentiation rules
Main lesson
The product and quotient rules from a table
Use this for “f(2) = 3, f′(2) = −1, g(2) = 4, g′(2) = 5. Find (fg)′(2) and (f/g)′(2).”
The chain rule on a power
Use this for “Find f′(1) for f(x) = (2x + 1)³.”
The derivative of A sin(bx)
Use this for “Find the slope of y = 4 sin(2x) at x = π/3.”
The derivative of A·eᵏˣ
Use this for “Find the slope of y = 3e0.5x at x = 2.”
Implicit slope on a circle
Use this for “Find dy/dx on x² + y² = 100 at (6, −8), and the tangent line there.”
Linear approximation of √x
Use this for “Use the linearization of √x at a = 25 to estimate √26.”
L’Hôpital’s rule on (ekx − 1) ÷ (mx)
Use this for “Find the limit of (e⁴ˣ − 1) ÷ (5x) as x → 0.”
Related rates and optimization
Main lesson
The largest field along a river
Use this for “A gardener has 16 m of fence for a plot along a straight river, with no fence on the river side. What sides give the largest area?”
Highs and lows of a cubic
Use this for “Find the local maximum and minimum of f(x) = x³ − 6x² + 9x + 1, and where the graph changes how it bends.”
A sliding ladder
Use this for “A 10 m ladder leans on a wall and its foot slides away at 1 m/s. How fast is the top sliding down when the foot is 6 m from the wall?”
Two cars moving apart
Use this for “Two cars leave a crossroads, one east at 15 km/h and one north at 20 km/h. How fast is the gap between them growing when they are 12 km and 5 km out?”
Filling a cone tank
Use this for “Water pours at 0.03 m³/s into a cone tank, point down, 1.5 m in rim radius and 3 m deep. How fast does the level rise when the water is 1 m deep?”
The steps of an optimization problem
Use this for “A closed can must hold 1000 cm³. What radius and height use the least metal?”
Definite integrals and the Fundamental Theorem
Main lesson
Riemann sums
Use this for “Estimate the integral of x² from 1 to 4 with 6 right-endpoint rectangles, then find the error.”
The area function and the Fundamental Theorem, part 1
Use this for “F(x) is the integral of 3t − 2 from 1 to x. Find F(4) and F′(4).”
Average value of a function
Use this for “Find the average value of f(x) = 2x² on [1, 4], and the c in [1, 4] where f(c) equals it.”
Displacement and distance from velocity
Use this for “A particle moves with v(t) = 3t − 6 m/s for 0 ≤ t ≤ 4 s. Find its displacement and the total distance it travels.”
Area between a line and a parabola
Use this for “Find the area of the region between y = 2x + 3 and y = x².”
u-substitution
Mean value theorem and curve sketching
Linear approximation, Newton's method and L'Hôpital's rule
Antiderivatives and initial-value problems
Course test
See what you know: 10 questions on this course
New questions each time. Pass to mark it complete.