Grade 12 CalculusStudy Guide
Formulas · worked examples · practice problems with the answer key
Built to BC Calculus 12. Also covers Alberta Mathematics 31 and the calculus in Ontario's MCV4U.
About this study guideWho it’s for. Students working through Grade 12 Calculus, and the teachers, tutors and parents printing handouts for them. Print it, copy it, staple it: it’s free for classroom use, whole or by the page. How to use it. Each module teaches its formulas in order (equation, picture, and what every symbol means), then closes with a practice set. Work the problems with a pencil; the answer key waits at the back so honest work gets an honest check. The two-page formula sheet up front is the exam-day companion, and every formula carries its number (Calculus 12-1 onward) so your written work can cite exactly which one you used. Do it live. Every problem in here has a living version in the Exam Room at formula.expert/exams/grade-12-calculus: new numbers every attempt, hints that pop down when you want them, stars and streaks when you don’t need them. When a printed problem stumps you, the matching lesson will coach you through it, step by step, free and with no sign-up. Other courses’ study guides live at formula.expert/study-guides. FiguresThe formula sheetModule 1 Limits & ContinuityPieces of a function · Undoing sine · Closing in · Factor, cancel, substitute · Left, right, both · Where it ends up · No lifting the pencil · A root must be there · The Gap Inspection In BC Calculus 12 · Alberta Mathematics 31 · Ontario MCV4U Module 1 practice1. Pieces of a function: A function is defined in two pieces: f(x) = x² + 4 for x < 3, and f(x) = x² for x ≥ 3. Determine f(1). 2. Pieces of a function: A function is defined in two pieces: f(x) = 3x + 1 for x < 1, and f(x) = 2x for x ≥ 1. Determine f(1). 3. Undoing sine: An exam question asks for the exact value of arcsin(1/2), in radians. Determine arcsin(1/2). 4. Undoing sine: An exam question asks for the exact value of arccos(−√2/2), in radians. Determine arccos(−√2/2). 5. Closing in: A table of values for a function f near x = 2 is recorded. From the left: x = 1.9 gives 4.8, x = 1.99 gives 4.98, x = 1.999 gives 4.998. From the right: x = 2.001 gives 5.002, x = 2.01 gives 5.02, x = 2.1 gives 5.2. Also, f(2) = 5. Determine the limit of f(x) as x approaches 2. 6. Closing in: The graph of a function g is the line y = 3x − 2 with a hole where x = 4; g(4) is undefined. Determine the limit of g(x) as x approaches 4. 7. Factor, cancel, substitute: Consider the limit of (x² − x − 2) / (x − 2) as x approaches 2. Determine the limit, if it exists. 8. Factor, cancel, substitute: Consider the limit of (x² − 36) / (x − 6) as x approaches 6. Determine the limit, if it exists. 9. Left, right, both: A function is defined in two pieces: f(x) = −3x − 4 for x ≤ 0, and f(x) = −x − 4 for x > 0. Determine the one-sided limits of f at x = 0, and whether the two-sided limit exists. 10. Left, right, both: Let g(x) = 2|x + 1| / (x + 1). Determine the one-sided limits of g at x = −1, and whether the two-sided limit exists. 11. Where it ends up: Let f(x) = (2x + 2) / (x³ + 2x − 5). Determine the limit of f(x) as x → ∞. 12. Where it ends up: Let f(x) = 2 + 7e⁻ˣ. Determine the limit of f(x) as x → ∞. 13. The Gap Inspection: No calculator. A function is defined in three lines: f(x) = x² − 2 for x < 3; f(3) = 5; f(x) = −x + 10 for x > 3. Inspect f at x = 3: evaluate each piece, find the one-sided limits, decide whether f is continuous there, and determine the value that would mend any gap. Module 2 The DerivativeFrom secant to tangent · The limit definition · Four ways to write it · The power rule · The tangent line · Rates of rates · Speed at an instant · The Speed Trap In BC Calculus 12 · Alberta Mathematics 31 · Ontario MCV4U Module 2 practice14. From secant to tangent: Let f(x) = 2x² − x − 2. The slopes of the secant lines through (−2, f(−2)) and (−2 + h, f(−2 + h)) are recorded as h shrinks: h = 1 gives −7, h = 0.5 gives −8, h = 0.1 gives −8.8, h = 0.01 gives −8.98. Determine the slope of the tangent line to the graph of f at x = −2. 15. From secant to tangent: Let f(x) = −x² − 2x − 4. Determine the slope of the secant line through the points on the graph of f where x = 0 and x = 0.5. 16. The limit definition: Let f(x) = −2x² + x. Determine f′(1) from first principles. 17. The limit definition: Let f(x) = 4x − 3. Determine f′(−3) from first principles. 18. The power rule: Let f(x) = −2x² + 4x + 3. Determine f′(2). 19. The power rule: Let f(x) = 10√x. Determine f′(25). 20. The tangent line: Let f(x) = x³ − 3x. Determine the y-coordinate of the point on the tangent line to f at x = 2 where x = 4. 21. The tangent line: Let f(x) = √x. Use the tangent line to f at x = 16 to estimate √15.2. 22. Rates of rates: Let f(x) = 2x³ − x² − 4. Determine f″(3). 23. Rates of rates: A particle moves along a straight line with position s(t) = 2t² + 6t + 2 metres after t seconds. Determine its acceleration at t = 4 s. 24. Speed at an instant: An object moves along a straight line with position s(t) = t² + 8t − 3 metres after t seconds. Determine its velocity at t = 2 s. 25. Speed at an instant: A ball is thrown straight up from a height of 5 m with an initial velocity of 15 m/s. Its height after t seconds is h(t) = −4.905t² + 15t + 5 metres, with up taken as positive. Determine the ball's velocity at t = 3 s. 26. The Speed Trap: No calculator. A cart moves along a straight track with position s(t) = 2t³ − 3t² + 4t + 3 metres after t seconds. Inspect the motion at t = 2 s: find the position, the velocity and the acceleration there, then use the tangent line to the position graph at that moment to predict where the line says the cart would be one second later. Module 3 Differentiation RulesThe product rule · The quotient rule · The chain rule · Exponentials and logs · Trig derivatives · Implicit differentiation · Which rule first · The Rule Book In BC Calculus 12 · Alberta Mathematics 31 · Ontario MCV4U Module 3 practice27. The product rule: Let f(x) = (x² − 2)(x³ − 2). Determine f′(−2). 28. The product rule: Two functions f and g are differentiable at x = 3, with f(3) = −5, f′(3) = 3, g(3) = 3 and g′(3) = 4. Determine the derivative of the product f·g at x = 3. 29. The quotient rule: Let f(x) = x³/(x² + 1). Determine f′(3). 30. The quotient rule: Two functions f and g are differentiable at x = 1, with f(1) = 4, f′(1) = 3, g(1) = −1 and g′(1) = −3. Determine the derivative of the quotient f/g at x = 1. 31. The chain rule: Let f(x) = (−x − 1)⁴. Determine f′(−2). 32. The chain rule: Let f(x) = √(x² + 15). Determine f′(7). 33. Exponentials and logs: Let f(x) = 4e⁻²ˣ. Determine f′(0). 34. Exponentials and logs: Let f(x) = −2x + 4 ln x. Determine f′(4). 35. Trig derivatives: Let f(x) = 5 sin x, with x in radians. Determine f′(5π/3). 36. Trig derivatives: Let f(x) = 4 cos(3x), with x in radians. Determine f′(5π/18). 37. Implicit differentiation: The point (4, 8) lies on the circle x² + y² = 80. Calculate dy/dx at (4, 8). 38. Implicit differentiation: The point (−3, 2) lies on the hyperbola x² − 2y² = 1. Calculate dy/dx at (−3, 2). 39. Which rule first: Let f(x) = (2x + 1)³(x − 1). Determine f′(0). 40. Which rule first: Let f(x) = x/(x² + 1). Determine f′(3). 41. The Rule Book: No calculator. Let f(x) = (x + 1)(2x − 3)². Open the rule book on f at x = 1: name the rules that f′(x) needs, find f(1) and f′(1), and use the tangent line at x = 1 to find where the line says y is at x = 2. Answer key
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