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Calculus solver

Limits, derivatives, integrals and the word problems built on them.

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  • Arithmetic to calculus

Calculus is two ideas and a bridge between them. The derivative measures how fast something changes; the integral adds up infinitely many small pieces; the fundamental theorem says each undoes the other. Nearly every exam question is one of those, dressed up in a context.

Naming the question first

Most lost marks come from starting the wrong machine, not from bad arithmetic. Identify the shape before you differentiate anything.

  1. Rate of change, slope, tangent, velocity

    That is a derivative. Decide which rule applies: product, quotient or chain, and often more than one.

  2. Area, total, accumulated, displacement, average value

    That is an integral. Definite if it has limits, indefinite if it needs a + C.

  3. Behaviour near a point, or at infinity

    That is a limit. Substitute first; only reach for factoring, conjugates or L’Hôpital when substitution gives 0/0 or ∞/∞.

  4. Two quantities changing together

    That is related rates. Write the equation connecting them, differentiate both sides with respect to time, then substitute the numbers last.

  5. Maximum or minimum

    That is optimisation. Set the derivative to zero, solve, and test whether each point is a maximum or a minimum.

Worked examples

Each one is solved the way the solver solves it: the rule first, then the line.

Find the derivative of (2x + 1)^5

ddx (2x+1)5\frac{d}{dx}\,(2x + 1)^5
  1. Chain rule: outside first
    5(2x+1)4⋅ddx(2x+1)5(2x + 1)^4 \cdot \frac{d}{dx}(2x + 1)
  2. Derivative of the inside
    ddx(2x+1)=2\frac{d}{dx}(2x + 1) = 2
  3. Multiply
    10(2x+1)410(2x + 1)^4
Answer
10(2x+1)410(2x + 1)^4

Evaluate the integral of (2x + 1) dx from 0 to 3

∫03(2x+1) dx\int_0^3 (2x + 1)\,dx
  1. Antiderivative term by term
    ∫(2x+1) dx=x2+x\int (2x + 1)\,dx = x^2 + x
  2. Fundamental theorem of calculus
    [x2+x]03=(9+3)−(0+0)\Big[ x^2 + x \Big]_0^3 = (9 + 3) - (0 + 0)
  3. Evaluate
    1212
Answer
1212

Find the limit as x approaches 0 of sin(x)/x

lim⁡x→0sin⁡xx\lim_{x \to 0} \frac{\sin x}{x}
  1. Substitution gives an indeterminate form
    sin⁡00=00\frac{\sin 0}{0} = \frac{0}{0}
  2. This is a standard limit
    lim⁡x→0sin⁡xx=1\lim_{x \to 0} \frac{\sin x}{x} = 1
  3. Or by L’Hôpital’s rule
    lim⁡x→0cos⁡x1=cos⁡0=1\lim_{x \to 0} \frac{\cos x}{1} = \cos 0 = 1
Answer
11

A 10 ft ladder leans against a wall. The bottom slides away at 2 ft/s. How fast is the top sliding down when the bottom is 6 ft from the wall?

x2+y2=102x^2 + y^2 = 10^2
  1. Relate the quantities
    x2+y2=100x^2 + y^2 = 100
  2. Differentiate both sides with respect to t
    2xdxdt+2ydydt=02x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0
  3. Find y when x = 6
    y=100−36=8y = \sqrt{100 - 36} = 8
  4. Substitute the known values
    2(6)(2)+2(8)dydt=02(6)(2) + 2(8)\frac{dy}{dt} = 0
  5. Solve
    dydt=−2416=−1.5\frac{dy}{dt} = -\frac{24}{16} = -1.5
Answer
dydt=−1.5 ft/s\frac{dy}{dt} = -1.5\ \text{ft/s}

The minus sign is the answer to "how fast is it falling": the height is decreasing at 1.5 ft/s.

Where marks get lost

Substituting numbers too early in related rates

Differentiate the general relationship first, then put the numbers in. Substituting first freezes the very quantity that is supposed to be changing.

Losing the + C

An indefinite integral is a family of functions. Without + C the answer is one member of it, and in a differential equation that costs the whole question.

Reaching for L’Hôpital too fast

It only applies to 0/0 and ∞/∞. Used on anything else it produces a confident wrong answer.

Questions

What is calculus?

The mathematics of change. Differential calculus measures how fast something changes, integral calculus adds up those changes, and the two are inverses of each other.

How do I find critical points?

Differentiate, set the derivative to zero, and solve; also include any point where the derivative does not exist. These are the candidates for maxima and minima.

How do I tell a maximum from a minimum?

Use the second derivative at the critical point. Negative means a maximum, positive means a minimum, and zero is inconclusive, so fall back to checking the sign of the first derivative on each side.

What is calculus used for?

Anything where a rate matters: velocity and acceleration, optimisation, growth and decay, areas and volumes, probability densities, and almost all of physics and engineering.

What is a Taylor series?

A way of writing a function as an infinite polynomial built from its derivatives at one point. Truncating it gives a polynomial that closely matches the function near that point.

How do I do related rates?

Write an equation relating the quantities, differentiate both sides with respect to time, then substitute the values you know. Substitute only after differentiating, never before.

What is the fundamental theorem of calculus?

It ties the two halves together: integrating a rate of change over an interval gives the total change, so a definite integral can be evaluated from an antiderivative.

What is the difference between Calculus 1, 2 and 3?

Calculus 1 is limits and derivatives, Calculus 2 is integration techniques and series, and Calculus 3 is the same ideas in several variables. The solver handles all three.