You can run a derivative backwards, remember the constant C, and use a starting value to pin C down.
Weeks 13-14
Calculus I course units
Work in order for a complete course, or jump directly to the unit you need for an upcoming assessment.
01
Limits and continuity
Weeks 1-2
You can find what a function approaches near a point, from the left or the right, and say where it breaks. You can also handle a limit as x runs off to infinity.
Factor and cancel to clear a 0/0 form.
Compare the two one-sided limits to test a piecewise function for continuity.
Find the limit of (3x squared + 5)/(2x squared - x) as x grows without bound.
Worked example
Find the limit of (x squared - 4)/(x - 2) as x approaches 2
Substituting gives 0/0
Factor: (x - 2)(x + 2)/(x - 2) cancels to x + 2
Answer4
Most common mistakeAnswering that the limit does not exist because x = 2 is outside the domain: the graph has a single hole at height 4 and the limit is 4.
You can build a derivative from the limit of a difference quotient and read it as a slope or a rate. You can also say where a function has no derivative at all.
Compute the derivative of x squared straight from the limit definition.
Read f'(3) = -2 as falling 2 units for every unit of x at that point.
Explain why the absolute value function has no derivative at zero.
Worked example
Use the definition to find the derivative of f(x) = x squared
[(x + h) squared - x squared]/h = (2xh + h squared)/h
Cancel one h: 2x + h
AnswerLet h shrink to 0: the derivative is 2x
Most common mistakeSetting h = 0 before cancelling: the quotient reads 0/0 and the derivative looks undefined, when cancelling first gives 2x.
You can differentiate products, quotients and nested functions quickly, without going back to limits. You can also look at a formula and name the rule it needs.
Apply the chain rule to (3x + 1) to the fifth power.
Use the quotient rule on x divided by (x squared + 1).
Differentiate sin x, e to the x, and ln x from memory.
Worked example
Differentiate y = (3x + 1) to the fifth power
Outside first: 5(3x + 1) to the fourth
Times the derivative of the inside, which is 3
Answery' = 15(3x + 1) to the fourth
Most common mistakeSkipping the inside derivative: writing 5(3x + 1) to the fourth and losing the factor of 3, so every slope comes out three times too small.
You can find a slope on a curve where y is tangled up with x, such as a circle. You can also write the tangent line at a listed point on that curve.
Differentiate x squared + y squared = 25 and solve for dy/dx.
Attach dy/dx every single time you differentiate a y term.
Write the tangent line at a point on an implicit curve.
Worked example
Find dy/dx for x squared + y squared = 25 at the point (3, 4)
Differentiate both sides: 2x + 2y(dy/dx) = 0
dy/dx = -x/y
Answer-3/4
Most common mistakeDifferentiating y squared as 2y with no dy/dx attached: that gives dy/dx = -x and a slope of -3 instead of -0.75, four times too steep.
You can link two changing quantities and find how fast one moves when you know the other. You can also set the equation up properly before any numbers go in.
Write an equation linking the quantities, then differentiate with respect to time.
Substitute the given instant only after differentiating.
Attach the right units, such as cubic cm per second.
Worked example
A balloon's radius grows 2 cm per second. How fast is its volume growing when r = 5 cm?
V = (4/3) pi r cubed, so dV/dt = 4 pi r squared times dr/dt
dV/dt = 4 pi (25)(2)
Answer200 pi, about 628.3 cubic cm per second
Most common mistakePlugging r = 5 in before differentiating: the volume becomes the fixed number 500 pi / 3, whose derivative is 0, so the balloon looks frozen.
You can find the largest or smallest value a quantity can reach, checking both critical points and endpoints. You can also justify that your answer really is the maximum.
Rewrite the goal in one variable using the constraint.
Solve f'(x) = 0 and test the sign of f' on both sides of the answer.
Check the endpoints of a closed interval as well as the interior.
Worked example
Split 20 into two numbers whose product is as large as possible
x + y = 20, so the product is P = x(20 - x) = 20x - x squared
P' = 20 - 2x = 0 gives x = 10, and P' changes from + to -, so it is a maximum
Answer10 and 10, with product 100
Most common mistakeSetting the constraint to zero instead of the derivative: solving 20 - x = 0 gives x = 20, whose partner is 0 and whose product is 0.
You can run a derivative backwards, remember the constant C, and use a starting value to pin C down. You can also turn a velocity function into a position function.
Find the antiderivative of 6x squared + 4.
Use a starting condition such as f(0) = 5 to find C.
Recover position from velocity and an initial position.
Worked example
Find f(x) if f'(x) = 6x squared + 4 and f(0) = 5
Antidifferentiate: f(x) = 2x cubed + 4x + C
f(0) = C, and we are told f(0) = 5
Answerf(x) = 2x cubed + 4x + 5
Most common mistakeLeaving off the + C: reporting f(x) = 2x cubed + 4x, which gives f(0) = 0 and contradicts the stated starting value of 5.
Before timing yourself, check whether you can explain Limits and continuity from a blank page. Then connect it to Derivative definition. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.
Use tools to verify, not to choose the method for you
The Calculus calculator can test calculations and representations used in Calculus I. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.
Know when to move to the full test
Move from Calculus I practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.
Before and after the syllabus
Learn the ideas, then practise them
The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.
Start with Limits and continuity if you are following the full sequence. If that unit feels automatic, open the Calculus I practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.
How do I know I am ready for the course test?
You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete Calculus I test only after you can correct practice errors from a blank page.
Which calculator supports this course?
The Calculus calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.
