1. Math 34A Chapter 5 examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

2. Percentage Error Sometimes you have an estimate for a number that is different from the actual value. It is useful to know how far off your estimate is. We have a formula for this: Percentage Error = Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

3. Percentage Error Sometimes you have an estimate for a number that is different from the actual value. It is useful to know how far off your estimate is. We have a formula for this: Percentage Error = Example: Find an estimate for the square root of 13. Just make a good guess, and try to be as accurate as you can. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

4. Percentage Error Sometimes you have an estimate for a number that is different from the actual value. It is useful to know how far off your estimate is. We have a formula for this: Percentage Error = Example: Find an estimate for the square root of 13. Just make a good guess, and try to be as accurate as you can. Suppose we choose 3.5 as our estimate. How accurate is this? Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

5. Percentage Error Sometimes you have an estimate for a number that is different from the actual value. It is useful to know how far off your estimate is. We have a formula for this: Percentage Error = Example: Find an estimate for the square root of 13. Just make a good guess, and try to be as accurate as you can. Suppose we choose 3.5 as our estimate. How accurate is this? The actual value is about 3.60555 (using a calculator). Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

6. Percentage Error Sometimes you have an estimate for a number that is different from the actual value. It is useful to know how far off your estimate is. We have a formula for this: Percentage Error = Example: Find an estimate for the square root of 13. Just make a good guess, and try to be as accurate as you can. Suppose we choose 3.5 as our estimate. How accurate is this? The actual value is about 3.60555 (using a calculator). Here is the calculation: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

7. Percentage Error Sometimes you have an estimate for a number that is different from the actual value. It is useful to know how far off your estimate is. We have a formula for this: Percentage Error = Example: Find an estimate for the square root of 13. Just make a good guess, and try to be as accurate as you can. Suppose we choose 3.5 as our estimate. How accurate is this? The actual value is about 3.60555 (using a calculator). Here is the calculation: This is the percent error in our estimate Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

8. The Change in f(x) In calculus we will often be concerned with finding rates of change. This is something that is very familiar, but the wording and notation can be confusing. Here are a few examples: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

9. The Change in f(x) In calculus we will often be concerned with finding rates of change. This is something that is very familiar, but the wording and notation can be confusing. Here are a few examples: Suppose f(x) = 3x+7. Find the change in f(x) when x changes from 2 to 5. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

10. The Change in f(x) In calculus we will often be concerned with finding rates of change. This is something that is very familiar, but the wording and notation can be confusing. Here are a few examples: Suppose f(x) = 3x+7. Find the change in f(x) when x changes from 2 to 5. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB f(2) = 3(2)+7 = 13 f(5) = 3(5)+7 = 22 To get the change in f(x) simply subtract the function values: f(5) – f(2) = 22-13 = 7

11. The Change in f(x) In calculus we will often be concerned with finding rates of change. This is something that is very familiar, but the wording and notation can be confusing. Here are a few examples: Suppose f(x) = 3x+7. Find the change in f(x) when x changes from 2 to 5. Suppose f(x) = 3x+7. Find the rate of change of f(x) when x changes from 2 to 5. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB f(2) = 3(2)+7 = 13 f(5) = 3(5)+7 = 22 To get the change in f(x) simply subtract the function values: f(5) – f(2) = 22-13 = 7

12. The Change in f(x) In calculus we will often be concerned with finding rates of change. This is something that is very familiar, but the wording and notation can be confusing. Here are a few examples: Suppose f(x) = 3x+7. Find the change in f(x) when x changes from 2 to 5. Suppose f(x) = 3x+7. Find the rate of change of f(x) when x changes from 2 to 5. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB f(2) = 3(2)+7 = 13 f(5) = 3(5)+7 = 22 To get the change in f(x) simply subtract the function values: f(5) – f(2) = 22-13 = 7 To get the rate of change in f(x) simply divide the change in f(x) by the change in x. This is just the slope of the line joining the 2 points.

13. The Change in f(x) Now let’s try it with a (slightly) harder function, and with variables. Suppose f(x) = x 2 +4 Find the change in f(x) when x changes from a to a+h. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

14. The Change in f(x) Now let’s try it with a (slightly) harder function, and with variables. Suppose f(x) = x 2 +4 Find the change in f(x) when x changes from a to a+h. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB f(a) = a 2 +4 f(a+h) = (a+h) 2 +4 To get the change in f(x) simply subtract the function values: f(a+h) – f(a) = [(a+h) 2 +4] – [a 2 +4] You can simplify this by multiplying it out and collecting like terms. f(a+h) – f(a) = [a 2 + 2ah + h 2 +4] – [a2+4] = 2ah + h 2

15. The Change in f(x) Now let’s try it with a (slightly) harder function, and with variables. Suppose f(x) = x 2 +4 Find the rate of change of f(x) when x changes from a to a+h. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

16. The Change in f(x) Now let’s try it with a (slightly) harder function, and with variables. Suppose f(x) = x 2 +4 Find the rate of change of f(x) when x changes from a to a+h. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB To get the rate of change in f(x) simply divide the change in f(x) by the change in x. This is just the slope of the line joining the 2 points. When we start doing calculus, we sill have one last step – taking the limit of this expression as h approaches 0. This will be called the derivative, and it will represent the slope of the tangent line to f(x) at the point where x=a. In this case our derivative would be 2a.

17. Summation Notation When you see the Greek letter Sigma (Σ) it means “sum”. It gives us a convenient shorthand way of writing a long sum of terms. Here are a few examples: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB You can also combine sums together, but be careful – the expressions inside must match. This one works fine, just connect them together to get one big sum. This one makes no sense. You can’t figure out what to put inside the sum.

18. Limits When you find a limit, you are asking what happens to a function when the variable approaches a certain number (from both sides). Here are some examples: For this one we have a well-behaved function, so just putting x=2 in the formula gives us the limit. If nothing goes wrong when you plug in the number, you probably have your limit. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB This time when you try to plug in x=2, you run into trouble. You get a value of 1/0. That is undefined. The limit does not exist. If you graph this function there is a vertical asymptote at x-2. This time when you try to plug in x=2, you run into trouble again. You get a value of 0/0. That is undefined, but we still have some hope. Factor out the top and you can get a simpler expression: The graph of this one would be a straight line (specifically y=x+2), but with a hole in the graph at the point (2,4).