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Calculus Toolkit - 8 Essential Tools for Analyzing Functions

A comprehensive guide to mastering calculus with 8 essential tools: finding zeros, intersections, even/odd functions, domains, asymptotes, tangents, derivatives, and approximations.

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Calculus Toolkit - 8 Essential Tools for Analyzing Functions

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  1. When you see… You think… A1. Find the zeros

  2. A1 To find the zeros...

  3. When you see… A2. Find intersection of f(x) and g(x) You think…

  4. A2 To find the zeros...

  5. When you see… A3 Show that f(x) is even You think…

  6. A3 Even function

  7. When you see… A4Show that f(x) is odd You think…

  8. A4 Odd function

  9. When you see… A5Find the domain of f(x) You think…

  10. A5 Find the domain of f(x) • Assume domain is (-∞,∞). • Restrictable domains: • Denominators ≠ 0 • Square roots of only non negative #s • log or ln of only positive #s

  11. When you see… A6Find vertical asymptotes of f(x) You think…

  12. A6 Find vertical asymptotes of f(x) Express f(x) as fraction, with numerator, denominator in factored form. Reduce if possible. Then set denominator = 0

  13. When you see… A7If continuous function f(x) has f(a) < k and f(b) > k, explain why there must be a value c such that a<c<b and f(c) = k. You think…

  14. A7 Find f(c) = k where a<c<b This is the Intermediate Value Theorem. We usually use it to find zeros between positive and negative function values, but it could be used to find any y-value between f(a) and f(b).

  15. When you see… B1 Find You think…

  16. B1 Find Step 1: Find f(a). If zero in denom, step 2 Step 2: Factor numerator, denominator and reduce if possible. Go to step 1. If still zero in denom, check 1-sided limits. If both + or – infinity, that is your answer. If not, limit does not exist (DNE)

  17. When you see… B2 Find where f(x) is a piecewise function. You think…

  18. B2 Show exists(Piecewise) Check 1-sided limits . . . .

  19. B3 When you see… Show that f(x) is continuous You think…

  20. .B3f(x) is continuous 1) exists 2) exists 3)

  21. B4 When you see… Find You think…

  22. B4 FindExpress f(x) as a fraction, determine highest power. If in denominator, limit = 0If in numerator, lim = +

  23. When you see… B-5Find horizontal asymptotes of f(x) You think…

  24. B5 Find horizontal asymptotes of f(x) Find and

  25. When you see… C1 Find f ’(x) by definition You think…

  26. C1Find f ‘( x) by definition

  27. When you see… C2Find the average rate of change of f(x) at[a, b] You think…

  28. C2 Average rate of change of f(x) Find f (b) - f ( a) b - a

  29. When you see… C3Find the instantaneous rate of change of f(x) ata You think…

  30. C3 Instantaneous rate of change of f(x) Find f ‘ ( a)

  31. When you see… C4 Given a chart of x and f(x) on selected values between a and b, estimate where c is between aandb. You think…

  32. C4 Estimating f’(c) between a and b Straddle c, using a value of k greater than c and a value h less than c. So

  33. When you see… C5 Find equation of the line tangent to f(x) at(x1,y1) You think…

  34. C5 Equation of the tangent line Find slope m = f ’(x). Use point (x1 , y1) Use Point Slope Equation: y – y1 = x – x1

  35. When you see… C6 Find equation of the line normal to f(x) at(a, b) You think…

  36. C6 Equation of the normal line

  37. When you see… C7Find x-values where the tangent line to f(x) is horizontal You think…

  38. C7 Horizontal tangent line

  39. When you see… C8Find x-values where the tangent line to f(x) is vertical You think…

  40. C8 Vertical tangent line to f(x) Write f ’(x) as a fraction. Set the denominator equal to zero.

  41. When you see… C9Approximate the value of f (x1 + a) if you know the function goes through (x1 , y1) You think…

  42. C9 Approximate the value of (x1 + a) Find the equation of the tangent line tof using y-y1 = m(x-x1). Now evaluate at x = x1+a. Note: The closer to a is to x1, the better the approximation. Note: Can use f’’, concavity to tell if it is an under- or overestimate.

  43. When you see… C10 Find the derivative of f(g(x)) You think…

  44. C10 Find the derivative of f(g(x)) Composition of functions! Chain Rule! f’(g(x)) ·g’(x)

  45. When you see… C11The line y = mx + b is tangent to f(x) at (x1, y1) You think…

  46. C11 y = mx+b is tangent to f(x) at (a,b) Two relationships are true: • The function and the line have the same slope at x1: (m=f ’(x)) • The function and line have same y-value at x1

  47. When you see… C12Find the derivative of g(x), the inverse to f (x) at x = a You think…

  48. C12 Derivative of g(x), the inverse of f(x) at x=a On g use (a, Q) On f use (Q, a) Find Q-value So

  49. C12 Derivative of g(x), the inverse of f(x) at x=a Interchange x with y. Plug your x value into the inverse relation and solve for y Solve for implicitly (in terms of y) Finally plug that y into

  50. When you see… C13 Show that a piecewise function is differentiable at the point awhere the function rule splits You think…

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