First in-class exam review This exam is over the material in Sections 2.1 2.6 and 3.1. Study this review, the material for quiz 1, quiz 2 and the appropriate homework assignments. My recommendation: Solve each problem on paper. Then check your answers against the solutions. I. Preliminary skills 1. Suppose f(x) = x 2 x and g(x) = 1 x. (a) What is f g? (b) What is g f? (c) What is the domain of g f? 2. (a) What is domain of f(x) = x 2 3? (b) What is the range of f(x) = x 2 3? 3. What is the domain of f(x) = 1 x 2 4? 4. What s the domain of f(x) = 5. What s the domain of f(x) = (x 2) x 2 4x + 3? (x + 2) x 2 4x + 4? 6. The length of a rectangle l is 3 times its width w. (a) What is the area of the rectangle as a function of w? (b) What is its perimeter as a function of w? 7. A rectangular garden is enclosed by a 120 ft rope. What is the area of the garden as a function of its side that is facing the road? 8. Area of a rectangular bulletin board is 100ft 2. What is the perimeter of the board as a function of height? 9. The cost and revenue function for a certain firm are given by C(x) = 12x + 2000 and R(x) = 20x, respectively. Find the company s profit function. 10. The monthly demand and supply function for Luminar desk lamp are given by p = d(x) = 1.1x 2 + 1.5x + 40 p = s(x) = 0.1x 2 + 0.5x + 15 respectively, where p is measured in dollars and x is units of thousands. Find the equilibrium quantity and price.
11. Suppose a rectangular package that has square cross section of x in x in is to have a combined length and girth of exactly 108 in. Find a function in terms of x giving the volume of package. ( Hint: The length plus girth is 4x + h where h is the length of the package.) x x h II. Limits 1. Find lim x 2 9 x + 3 2. Find lim x 2 9 x 3 3. Find lim x 2 + 9 x 3 4. Find lim x 2 8 x + 3 5x 20 + 3 5. Find lim x 2x 17 + 3 5x 17 + 3 6. Find lim x 2x 20 + 3 5x 20 + 3 7. Find lim x 2x 20 + 3 8. Find lim x 1 5x 20 + 3 x 20 1 Page 2
x if x < 1 9. Suppose f(x) = x + 1 if 1 x < 2 x 2 1 if 2 x (a) What s lim f(x)? x 1 (b) What s lim f(x)? x 1 + (c) What s f(1)? (d) Is f continuous at x = 1? (e) Is f continuous at x = 2? 10. If, t months after you join twitter, you have will you have? 100t 2 + 7 2t 2 followers, in the long run how many followers + 50t + 1 11. Consider the following graph of a function y = f(x): 6. 5. 4. 3. f(x) 2. 1. 3. 2. 1. 0 1. 2. 3 4 5 6 7 8 1. 2. 3. (a) Is f continuous at x = 2? (b) Is f continuous at x = 1? (c) Is f continuous at x = 4? (d) Estimate lim f(x). x 2 (e) Estimate lim f(x). x 2 + (f) Estimate lim f(x). x 1 (g) Estimate f( 1). 12. (a) Use the Intermediate Value Theorem to show that for some a between -1 and 1 such that 5a 17 + 3a 5 = 3. (b) We know that f(x) = 1 x never equals 0. But f( 1) = 1 < 0 < 1 = f(1). Why can t we use the Intermediate Value Theorem to show that there is some x with 1 x = 0? Page 3
III. Derivative 1. Consider the function pictured in II.11. Does it have a derivative at x = 2? at x = 4? Why? 2. (a) Using the definition of derivative (a.k.a. the four-step method) find f (x) if f(x) = 2x 2 + x. (b) Find the slope of tangent line at x = 1. (c) Find the equation of tangent line at ( 1, 1). 3. Suppose f(x) = 4x 2 + 6x. Consider the graph of y = f(x). (a) Find the slope of the tangent line at x = 1. (b) Find the equation of the tangent line at the point ( 1, 2). IV. Applications of the derivative 1. The percentage probability of survival of cattle from a particularly deadly epidemic t weeks after diagnosis is P (t) = 50t 3.2 when 1 t 6 What s the rate of change (a.k.a. instantaneous rate of change) in the survival rate when t = 5? ( Round your answer to 4 decimal places.) 2. The distance (in centimeters) from its starting point by a snail moving back and forth in a straight line after t hours is s(t) = t 3 + 4t 2 + 15t, 0 t 5. (a) What is its average velocity during that time? (b) What is its instantaneous velocity at time t? (c) What is the distance traveled by snail after 2 hours. 3. The imaginary country of Positivityland has a GDP (= gross domestic product) calculated by the equation G(t) = 0.3t 3 + 3.1t 2 + 60, where t is the number of years since 2000. What is the rate of change of the GDP in 2016? Page 4
Short notes These notes are not going to appear on your exam. Finding the domain when the rule ( closed form) of the function is given. (a) Find zeros of the denominator. (b) Find the inequality that makes under the square root negative. (c) Draw a number line. Eliminate all points in part a and b from the number line. (d) What is left of the number line is in form of intervals. Write the intervals in the interval form and put a union in between them. Finding the domain and range using your calculator. Graph the function. All possible x-values is the domain. All possible y-values is the range. Some limit and continuity notes: (a) My recommendation: If you are having trouble answering any questions about a piece-wise defined function, graph the function first. (b) To find limit of polynomials, plug in the value. (c) To find the limits at infinity, divide numerator and denominator by x n where n is the degree of the denominator. So only terms with largest degree survive the limit and the rest will vanish. (d) 0 : If both numerator and denominators are polynomials, then they have common factors. Simplify 0 and plug in the value. (e) To graph a piece-wise defined function, divide your space into vertical strips which each strip width is the domain of one piece of function. Then graph each piece of function in the appropriate strip. Last, decide about open or closed circles ( end points). (f) lim f(x) is the limit from below. x a (g) lim f(x) is the limit from above. x a + (h) If f is defined around x = a ( it doesn t have to be defined at x = a.), then lim f(x) = x a lim f(x) = L if and only if lim f(x) = L. x a + x a (i) Continuity of Polynomial and Rational Functions i. A polynomial function y = P (x) is continuous at every value of x. ii. A rational function R(x) = p(x)/q(x) is continuous at every value of x where q(x) 0. (j) f is continuous at x = a if and only if (1)f(a) is defined, (2) lim f(x) exists and (3) lim f(x) = f(a). x a x a (k) If lim f(x) or lim f(x) or lim is infinity or negative infinity then the limit does not exists. x a + x a x a (l) The intermediate value theorem. Let f be continuous on [a, b] and M a value between f(a) and f(b), then there exist number r in [a, b] such that f(r) = M. (m) Existence of zero of a continuous function. Let f be continuous on [a, b] and f(a) and f(b) have different signs, then f has a zero in [a, b]. (n) How to find algebraic expression (rule) for function f g when algebraic expression for f and g is given. Write f with big parenthesis whenever you see x. Write g(x) inside those parenthesis. Four-step process for finding f (x) is (a) Compute f(x + h). Which is writing out f(x) and replacing x by (x + h) everywhere. (b) Form the difference f(x + h) f(x) f(x + h) f(x) (c) Form the quotient. h (d) Compute the limit f f(x + h) f(x) (x) = lim. h 0 h Average rate of change for function f over interval [a, a + h] is: f(a + h) f(a). h Page 5
The instantaneous rate of change of function f at a is: f(a + h) f(a) lim. ( This limit is also called derivative or f (a) at a.) h 0 h Notice the difference between Average and instantaneous rate of change. Differentiability and continuity theorem: If f is differentiable at x = a, then f is continuous at x = a. f is NOT differentiable at x = a if any of the following cases is true: (a) Graph of f has corners at x = a. (b) f is not continuous at x = a. (c) The tangent line is vertical at x = a. f (a) is the slope of tangent line to graph of y = f(x) at (a, f(a)). The equation of Tangent line at x = a to graph of y = f(x) can be found using the slope-point formula: y y 0 = m(x x 0) where m = f (a), x 0 = a and y 0 = f(a). Tangent line problems. When f(x) and x = a are given. (a) Find the derivative. f (x). (b) Plug in x value a in f (x). That is the slope of the tangent line. m = f(a) (c) Plug in a in original f(x) to get y-value f(a). (d) Use m and (a, f(a)) in the point-slope equation. y f(a) = m(x a). Horizontal tangent line if and only if f (a) = 0. To solve a word problem involving a geometrical shape. (a) Draw the geometrical shape. (b) Find a constraint. That is, a number or relation that is forced on the geometrical shape. You need this to solve all independent variables with respect to one. (c) Find the function that the problem is asking for. Replace the expressions of independent variable in the function. (d) What is the independent variable that was asked in the problem? Go back and use that to do the blue parts above. Page 6