Bridge to university · Mathematics
Bridge: getting ready for first-year calculus
Limits by intuition, slopes and rates, what a derivative means, and the algebra slips that cost marks in first-year calculus, with examples.
37 cards5 topicsAbout 15 minutes a roundPreparation for a typical first-year differential calculus course
About this deck
First-year calculus moves fast, and most students who struggle are not stuck on new ideas but on the algebra underneath them. This deck prepares you for the first month of a university calculus course. It builds intuition for limits and continuity, connects average rate of change to slopes of secant lines, explains what a derivative means in words and in units, previews the power rule, and shows how to read increasing, decreasing and concavity from a graph. A whole tag is devoted to algebra traps that markers see every term.
Use it in the summer before university, or alongside the first weeks of the course. If you took a Grade 12 calculus course you will recognize much of this; focus on the interpretation cards, since university exams often ask you to explain a result, not just compute it. If you did not take calculus in high school, read the slope and limit cards slowly and do every example by hand. Typical mistakes to watch for: treating 0/0 as an answer, expanding (a + b)^2 without the middle term, dividing both sides by a variable, writing f(x + h) as f(x) + h, and forgetting units when describing a rate of change.
Before you start: Grade 12 functions (Advanced Functions, Math 30-1, Pre-Calculus 12 or equivalent).
All 37 cards
Cover the definitions with a sheet of paper and test yourself, or use the study modes above.
Slopes and rates
- Slope of a line
- Rise over run, (y2 - y1)/(x2 - x1). A straight line changes at the same rate everywhere.Example: (1, 3) to (4, 12): 9/3 = 3
- Average rate of change
- Change in output divided by change in input over an interval: (f(b) - f(a))/(b - a). It is the slope of a secant line.Example: f(x) = x^2 from 1 to 3: (9 - 1)/2 = 4
- Secant line
- A line through two points on a curve; its slope gives the average rate between those points.Example: Joining (1, 1) and (3, 9) on y = x^2
- Tangent line
- The line that touches a curve at a point and matches its direction there; its slope is the instantaneous rate.Example: At x = 2 on y = x^2 it has slope 4
- Instantaneous rate of change
- What the average rate approaches as the interval shrinks toward zero width; it is computed with a limit.Example: A speedometer reading rather than a trip average
- Difference quotient
- (f(x + h) - f(x))/h: the average rate over a tiny interval of width h, and the starting point for every derivative.Example: For f(x) = x^2 it simplifies to 2x + h
Limits
- Limit (intuition)
- The value f(x) gets close to as x gets close to a, whether or not f(a) exists or matches it.Example: (x^2 - 1)/(x - 1) approaches 2 as x approaches 1, though it is undefined at 1
- One-sided limits
- Approaching a from the left only or from the right only. The two-sided version exists only when both agree.Example: For |x|/x at 0: left gives -1, right gives 1, so no two-sided value
- Limit at a hole
- When substitution gives 0/0, factor and cancel, then substitute. The graph has a gap but the approach value still exists.Example: (x^2 - 9)/(x - 3) = x + 3 for x not 3, so it approaches 6
- Indeterminate form 0/0
- A signal that direct substitution failed and more algebra is needed. It does not mean the answer is 0, 1 or undefined.Example: sin(x)/x gives 0/0 at 0 but approaches 1
- Vertical asymptote
- A line x = a near which outputs grow without bound, often where a denominator is 0 but the numerator is not.Example: 1/(x - 2) has one at x = 2
- Horizontal asymptote
- A height y = L that f(x) approaches as x becomes very large positive or very large negative.Example: (3x^2 + 1)/(x^2 - 5) approaches 3
- Continuity
- f has no break at a when f(a) exists, the limit as x approaches a exists, and the two are equal.Example: A hole, a jump or an asymptote each breaks it
- Rationalizing
- Multiplying by a conjugate over itself to clear a square root; a standard way past a 0/0 form.Example: (sqrt(x) - 2)/(x - 4) = 1/(sqrt(x) + 2), which approaches 1/4 as x approaches 4
Derivatives
- Derivative
- The instantaneous rate of change of f: the limit of (f(x + h) - f(x))/h as h approaches 0. Geometrically, the slope of the tangent.Example: For f(x) = x^2 the result is 2x
- Notations f'(x) and dy/dx
- Two ways to write the rate of change of y with respect to x. Treat dy/dx as one symbol, not a fraction to cancel.Example: If y = x^3, both equal 3x^2
- Units of a derivative
- Output units per input unit. Position in metres against time in seconds gives metres per second.Example: Cost in dollars against items made gives dollars per item
- Derivative as velocity
- If s(t) is position, s'(t) is velocity and the next one, s''(t), is acceleration.Example: s(t) = 5t^2 gives velocity 10t and acceleration 10
- Sign of the derivative
- f' > 0 on an interval means f rises there; f' < 0 means it falls; f' = 0 marks a possible peak, valley or flat spot.Example: f(x) = x^2 falls for x < 0 and rises for x > 0
- Power rule preview
- d/dx of x^n is n x^(n-1) for any real n. First-year courses justify it, then use it constantly.Example: d/dx of sqrt(x) = x^(1/2) is 1/(2 sqrt(x))
- Interpreting f'(3) = -2
- At x = 3 the output is decreasing at 2 output units per input unit.Example: If f(t) is the temperature in degrees t hours after noon, it is cooling at 2 degrees per hour at t = 3
- Rewriting before differentiating
- Turn roots and fractions into powers first: 1/x^3 = x^(-3) and the cube root of x = x^(1/3).Example: d/dx of 4/x^2 = d/dx of 4x^(-2) = -8x^(-3)
- Difference quotient for f(x) = 1/x
- (1/(x + h) - 1/x)/h = (x - (x + h))/(x h (x + h)) = -1/(x(x + h)), which approaches -1/x^2 as h approaches 0.
Algebra traps
- Trap: (a + b)^2
- It equals a^2 + 2ab + b^2, not a^2 + b^2. The middle term is the piece most often lost.Example: (x + 3)^2 = x^2 + 6x + 9
- Trap: sqrt(a^2 + b^2)
- Square roots do not split over addition, so this does not simplify to a + b.Example: sqrt(9 + 16) = 5, not 7
- Trap: dividing by a variable
- Dividing both sides by x can throw away the solution x = 0. Move everything to one side and factor.Example: x^2 = 3x becomes x(x - 3) = 0, so x = 0 or x = 3
- Trap: distributing a negative
- A minus sign in front of brackets changes the sign of every term inside.Example: 5 - (x - 4) = 5 - x + 4 = 9 - x
- Trap: fraction of a sum
- 1/(a + b) is not 1/a + 1/b; a sum in the denominator cannot be split.Example: 1/(2 + 2) = 1/4, but 1/2 + 1/2 = 1
- Trap: exponent on a coefficient
- (3x)^2 = 9x^2, while 3x^2 squares only the x. Brackets decide what the exponent covers.Example: At x = 2: (3x)^2 = 36 but 3x^2 = 12
- Trap: -x^2 when x = -3
- The exponent acts before the leading minus, so the value is -9. Write (-x)^2 only when the whole negative is squared.Example: On a calculator, type -(-3)^2 to check
- Trap: f(x + h)
- Replace every x in the rule with (x + h); the result is not f(x) + h.Example: f(x) = x^2 gives f(x + h) = x^2 + 2xh + h^2
Graph reading
- Reading slope sign from a graph
- Where the curve climbs from left to right the derivative is positive; at a smooth peak or valley it is zero.Example: A downhill stretch means a negative rate
- Concave up
- Shaped like a cup: slopes are increasing, so the second derivative is positive.Example: y = x^2 everywhere
- Corner or cusp
- A sharp point where no single tangent slope exists, so f is not differentiable there even if it is continuous.Example: y = |x| at x = 0
- Local maximum
- A point higher than all nearby points. For a smooth curve the slope there is 0 and changes from positive to negative.Example: y = -x^2 + 4 at x = 0
- Optimization idea
- Finding the largest or smallest value of a quantity such as area or cost by checking where its derivative is 0 and at the endpoints.Example: Largest rectangle with a fixed perimeter turns out to be a square
- Related rates idea
- Problems where two changing quantities are tied by an equation, so knowing how fast one changes tells you how fast the other does.Example: A growing circle: area and radius change together through A = pi r^2