Math 34A

1. Math 34A Final Review Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

2. 1) Use the graph of y=10x to find approximate values of a) 500.3 b) y’(0.65) solution for part a) first write an equation: x= 500.3 do the logarithm of both sides: log(x) = log(500.3) expand the right side using log rules: log(x) = 0.3log(50) log(x)= 0.3[log(5)+log(10)] log(x) = 0.3log(5)+0.3(1) now look up log(5) on the graph – it is about 0.7 log(x) = 0.3(0.7)+0.3 = 0.21+0.3 = 0.51 finally we take the antilog of both sides: x = antilog(0.51) = 100.51 Look this up on the graph: 100.51 ≈3.2 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

3. 1) Use the graph of y=10x to find approximate values of a) 500.3 b) y’(0.65) solution for part b) Draw the tangent line to the graph at x=0.65. Use the graph to find 2 points on the tangent line at 0.65 Calculate the slope as usual From the graph, it looks like we can use the points (0.4,2) and (.9,7) the slope should be about 10.3 (10 is close enough) Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

4. 2) Find values of x where the function f(x) = x4 – 4x3 + 4 a) is increasing b) is concave down c) has a minimum point We will need derivatives of f(x): f(x) will be concave down where f’’(x) is negative set f’’(x) = 0 to determine where f(x) might switch concavity from up to down, or vice-versa: f(x) will be increasing where f ’(x) is positive set f ’(x) = 0 to determine where f(x) might change sign: f ’ neg. f ’’ pos. f ’ neg. f ’’ neg. f ’ pos. f ’’ pos. Again, checking points in each region shows that f(x) is concave down for 0<x<2 The minimum point occurs when x=3. At this point, f ’(x)=0 and f ’’(x) is positive. 3 2 0 0 A number line can help – checking points in each region shows us that f(x) is increasing when x>3 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

5. 3) A function f(x) is graphed below. a) For what values of x is f(x)<0? 1.1<x<3.9 b) For what values of x is f ‘(x)<0? x<0 and 0<x<3 c) For what values of x is f “(x)>0? X<0 and x>2 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

6. 4) Given m(t) = 3e5t – k2π + t-2 a) Find m’(t) b) Find m’’(2) notice that k2π is constant, so its derivative is 0 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

7. 5) A theater currently sells tickets at a price of $8, and sells 1000 tickets. It is estimated that for each 10-cent increase in the price, 20 fewer tickets will be sold. What is the optimal price for the theater to charge (to maximize their income)? We want to maximize revenue, so we need a formula for revenue in terms of price. The basic formula is something like Revenue = (Price)(#of tickets sold) Define variables: f(x)=revenue when price is $x per ticket We will need a formula for #of tickets sold in terms of price, so define y=# of tickets sold. One way to do this type of problem is to realize that the formula for # of tickets will be linear. We already know one point on the line: (8,1000) – this is given in the first sentence. We can find another point on the line by using the given info about the price increases: For example, if the price is $9, 200 fewer tickets will be sold, so (9,800) will be on the line. Now we just need to use y=mx+b to find the equation of the line. The slope is using (8,1000) we get: This is our equation for the # of tickets. Putting this into our revenue equation: To find the maximum, we set f ’(x) = 0: The optimal price to charge is $6.50 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

8. 6) Given the two functions f(x) = 3x2 – 8x and g(x) = -2x2 + 5 a) Find the equation of the line tangent to f(x) at x=2 b) For which value of x is the slope of f(x) equal to the slope of g(x)? answer for a) The slope of the tangent line will be the derivative at x=2: We need a point on the line as well. Using x=2, we find f(2)=3(2)2-8(2)=-4. Thus (2,-4) is on the line. Now just use point-slope form for the equation of the line: answer for b) We need the first derivative of each function to be equal: Tangent line equation Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

9. 7) A rock is thrown straight upward from the ground. Its velocity (in meters per second) is given by the formula v(t) = 20 – 10t, where t is the time in seconds after the rock is thrown. a) When does the rock reach its highest point? b) What is the speed of the rock as it lands? For part a) you know that when the rock reaches its high point, the velocity will be zero. For part b) you can use a couple of different concepts. Option 1: The flight of the rock will be symmetrical. As long as it begins and ends at the same height, it will take the same amount of time on the way up, and on the way down. This gives a total flight time of 4 seconds. Find the velocity at t=4: So the speed will be 20m/sec. Speed is always positive. Option 2: The speed of the rock will be symmetrical. At the same height, it will have the same speed on the way up and on the way down (velocity is negative on the way down). Thus the speed when it lands will be the same as the speed when it is thrown. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

10. 8) The table gives the position of a car at various times during an afternoon drive. a) Find the average speed of the car between 2pm and 4pm. b) During what time interval was the average speed greatest? a) This is just a slope problem: b) For this part, just find the averages for each interval (same method as part a) The largest value is from 1:30 to 2pm. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

11. 9) A cylindrical container with no top is to be made to hold a volume of 12π cm3. Material for the side costs 2 cents per cm2 and the material for the bottom costs 3 cents per cm2. What are the dimensions of the least expensive container? We need formulas for the Volume and the Cost: 2πr r h side bottom h r Cost = (2)(area of side) + (3)(area of bottom) This has 2 variables, so we need to replace one of them using the volume equation: solve the volume formula for h: now substitute into the cost equation: We need to set the derivative of this cost function equal to zero to find the minimum: Radius = 2 cm Height = 3 cm plug this back in to find h: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB