APSC 171 – Calculus 1

1. APSC 171 – Calculus 1 Elizabeth Lutton (e.lutton@queensu.ca)

2. Overview Derivatives • Basic facts • Power, product, quotient, and chain rules • Implicit differentiation • Logarithmic differentiation • Linear approximations Integrals • Approximations (Reimann sums) • Solving using Fundamental Theorem of Calculus • Substitution • Improper integrals

3. Overview • Integration by parts • Partial fractions • Area between curves • Volume (slices and cylinders) • Workproblems • Slope fields • Differential equations Limits:L’Hôpital’s Rule Parametric Equations Last minute notes

4. Derivatives

5. Derivatives Derivatives to memorize: Memory tip: notice that derivatives of “co” functions are negative and “non-co” functions are positive

6. Derivatives More derivatives to memorize: Again notice the derivatives of “co” functions are negative and “non-co” functions are positive

7. Derivatives And more… Special cases because

8. Derivatives Power Rule: Example:

9. Derivatives Product Rule: “derivative of the first times the second, plus the first times the derivative of the second” Example: = ? = =

10. Derivatives Example: Simplify first! Write everything in terms of sine and cosine = = so we don’t even need the product rule =

11. Derivatives Quotient Rule: “derivative of the first times the second, MINUSthe first times the derivative of the second, all over the denominator squared” Compare to Product Rule: Product rule has + instead of – and no denominator

12. Derivatives Quotient Rule: Example: = =

13. Derivatives Tip: The Product Rule is easier to use than the Quotient Rule, so you can raise the denominator to an exponent of -1 and then use the Product Rule instead of the Quotient Rule Example: same as answer from previous slide

14. Derivatives Example: Simplify first! Factor the numerator = = = 2

15. Derivatives Moral of the story: Before you do anything, check if you can SIMPLIFY, SIMPLIFY, SIMPLIFY! Picture courtesy of http://www.zedge.net/profile/OoHumairahoO/

16. Derivatives Chain Rule (used for compositions of functions): f(x) is the “outer” function and g(x) is the “inner” function

17. Derivatives Example: = = = Tip: Rewriting square, cubic, etc. roots as numerical exponents when taking derivatives helps you to make fewer mistakes

18. Derivatives (Implicit) Implicit differentiation: what is it and when do you use it? • It’s used when y is a function of x instead of just another variable:x and y are related in some way • Legs and dogs vs. cats and dogs • Implicit differentiation is just a special case of the chain rule • 2y is a composition of functions where 2 is the “outer” function and y is the “inner” function

19. Derivatives (Implicit) Example: Do you use implicit differentiation or not? How do you know? • Find the tangent slopes of Yes, because y is obviously a function of x b) No, because we are not told that y depends on x in any way; y seems to be a variable not a function. Therefore the answer to this problem is zero.

20. Derivatives (Implicit) If in doubt, use implicit differentiation rather than not

21. Derivatives (Implicit) Example: Find the tangent slopes of (part a on previous slide) “tangent slopes” means derivatives!

22. Derivatives A note on notation: is a VARIABLE is a FUNCTION is the same as y’(x) means “derive the following with respect to x”

23. Derivatives (Implicit) Example (question 3 on 2008/2009 final): Calculate the tangent slope to the curve at the point (1,2). Note that you can substitute in values for x and y before solving for . It’s easier this way.

24. Derivatives (Implicit) Therefore the tangent slope to the curve at (1,2) is

25. Derivatives (Logarithmic) Logarithmic differentiation: when do you use it? • Use when the variable with which you are differentiating with respect to (usually x) is in the base AND the exponent • e.g.

26. Derivatives (Logarithmic) Example (question 2 on 2010/2011 final): Find the derivative when

27. Derivatives (Logarithmic) Be sure to write everything in terms of x (and not y) in your final answer for logarithmic differentiation questions • See last two steps of previous slide

28. Derivatives (Linear Approximation) Linear approximations use the y value f(a) at a point x=a, and the tangent line at x=a to approximate the value of a function

29. Derivatives (Linear Approximation) Slope of tangent line at x=a+Δx Slope of tangent line at x=a Tangent line to f(x) at x=a (slope = f’(a))

30. Derivatives (Linear Approximation) Memorize this equation:

31. Derivatives (Linear Approximation) Example: Approximate the value of 3+e0.2 0.2 is close to 0 so use a=0 Actual value:

32. Derivatives (Linear Approximation) Example: A circle has a radius of 12±0.3 mm. Estimate the maximum possible error in the area of the circle the maximum possible error in the area of the circle is mm max error is 0.3 absolute value because error is ±0.3

33. Optimization To solve optimization problems: • Translate the given word problem into a mathematical function f(x) • Find the derivative of the function and set it equal to zero to find the x values of the critical points • i.e. the critical points are the solutions to f’(x)=0 • Test the critical points using either the first or second derivative test to see if they are local maximums or minimums

34. Integrals

35. Integrals Integrals are “signed” or “net” area, which means they can have negative values +

36. Integrals The integral on the previous slide could have also been calculated as follows: Area of blue triangle – area of red triangle

37. Integrals (Approximations) Left Riemann sum: • Underestimate for increasing functions • Overestimate for decreasing functions http://www.vias.org/calculus/img/04_integration-10.gif

38. Integrals (Approximations) Right Riemann sum: • Underestimate for decreasing functions • Overestimate for increasing functions http://commons.wikimedia.org/wiki/File:Riemann-Sum-right-hand.png

39. Integrals (Approximations) Midpoint Riemann sum: • A better approximation than left and right Riemann sums http://www.ltcconline.net/greenl/courses/116/IntegrationApps/riemann.htm

40. Integrals (Approximations) Trapezoidal sum: • Uses trapezoids rather than rectangles to approximate the area under a curve • A better approximation than left, right, and midpoint approximations http://www2.seminolestate.edu/lvosbury/CalculusI_Folder/Calc%20I%20exam4.htm

41. Integrals (Approximations) Simpson’s Rule: • Uses pieces of parabola to approximate the area under a curve Memorize this formula http://eg1002.pbworks.com/w/page/6607488/Integration

42. Integrals Properties of Integrals: This is a rectangle where the width is zero so the area must be zero

43. Integrals More properties of integrals: where c is a constant where

44. Integrals What’s the difference between a definite and an indefinite integral? • A definite integral has bounds whereas an indefinite integral does not • e.g. is a definite integral in an indefinite integral

45. Integrals • When solving indefinite integrals, find the antiderivative of the function • where F(x) is the antiderivative of f(x) (i.e. F’(x) = f(x)) • Don’t forget the +C!!!!!!!!!!!! • e.g. where C is a constant specific to the function

46. Integrals Different functions with the same derivative (which is why we need the +C term)

47. Integrals • When solving definite integrals, use the second part of the Fundamental Theorem of Calculus • This theorem is also helpful for physics problems • e.g. where s(t) is the position function and v(t) is the velocity function • Recall and know this!

48. Integrals Example (question 2 on 2010/2011 final): = Tip: sometimes it may be easier to see an antiderivative if you use negative exponents and rewrite square roots as fractional exponents = =

49. Integrals = = =

50. Integrals (Substitution) Method of Substitution • Use when you see a function and its derivative in an integral • e.g. use substitution on because • If solving an indefinite integral, don’t forget to convert back to the original variable • i.e. change u back to x • If solving a definite integral, don’t forget to change the bounds of integration