Topics and Concepts. 1. Limits

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1 Topics and Concepts 1. Limits (a) Evaluating its (Know: it exists if and only if the it from the left is the same as the it from the right) (b) Infinite its (give rise to vertical asymptotes) (c) Limits at infinity (give rise to horizontal asymptotes) (d) Continuity (Know: it definition, types of discontinuities) 2. Differentiation (a) Limit definition and geometric reason for it (b) Differentiation formulas (product rule, quotient rule, chain rule) (c) Higher derivatives (d) Implicit differentiation (e) Finding equations of tangent lines (f) Reasons a function might fail to be differentiable (Know: differentiable implies continuous, but not the other way around) (g) Position, velocity, acceleration. Extreme Values (a) Statement of the extreme value theorem (b) Local vs Absolute maximum or minimum (c) Definition of a critical point and classifying it (d) Definition of concavity (using second derivative) and an inflection point (e) Shape of a graph as determined by the first and second derivatives 4. Integration (a) Definition of the integral via Riemann sums (b) Antiderivative (and definition of indefinite integral) (c) Fundamental Theorem of Calculus I and II (d) Substitution 1

2 Review problems on its and derivatives 1. Evaluate the following its. (x + 1) 2 1 (a) x 0 x 4x + 1 (b) x 2 x 2 ( 1 (c) x 0 x 1 x 2 + x ) (d) (e) (f) (g) x x 2 + e x 5e x + x 10 x x2 e x2 x 0 x2 ln x + x 0 x x + 2. Find the vertical asymptotes of f(x) = 1 x 2. Say whether the function approaches positive 1 or negative infinity from the right/left of the asymptote. 4x. Find the horizontal asymptotes of f(x) = State which asymptote f approaches as x 5 x and as x. sin x 4. Use the squeeze theorem to show that x x 2 = Find a and b such that f(x) is continuous, where x 2 + if x < 0 f(x) = ax + b if 0 x 2 x if x > The graph of a function f(x) is shown below. (a) List the values of x at which f is discontinuous. (b) List the values of x at which f is not differentiable. 7. Find f (x) if f(x) = sin(x 4 ). 8. Find y implicitly if cos(xy) = 1 + sin(y). You do not need to solve for it. 9. Find the equation of the tangent line to the curve x 2 + 2xy y 2 + x = 2 at the point (1, 2). 2

3 10. Find the absolute minimum and absolute maximum values of f(x) = (x 2 1) on [ 1, 2]. 11. Find and classify the critical points of the curve f(x) = x 4 2x Where is f increasing? Decreasing? Where is the f concave up? Concave down? 12. The graph of f(x) is shown below. (a) On what interval(s) is f (x) positive? Explain. (b) On what interval(s) is f (x) positive? Explain. 1. If F (x) is a function such that F (x) = 2 cos x and F (0) =, what is F (x)? 14. What is d dx x 2 e t sec t dt 15. What is d x sin(t 2 ) dt dx 0 Integration Practice: Chapter 5 Review, odd problems #9-19, #2-5 Concept Questions Disclaimer: If I ask for a definition or what something means or says I am looking for the mathematical definition/description. A description in words will only get partial credit. 1. Suppose that the graph of f is given. Write an equation for each of the graphs that are obtained from the graph of f as follows. (a) Shift 2 units upwards. (b) Shift units to the left. (c) Reflect about the x-axis. (d) Stretch vertically by a factor of 5. (e) Shrink horizontally by a factor of How arcsin x defined? What is its domain and range?. What does the Squeeze Theorem say?

4 4. What does it mean for f to be continuous at a? 5. State either definition of the derivative of f at x = a. Explain its meaning. 6. Name three distinct ways in which a function can fail to be differentiable at a. 7. Explain the difference between an absolute maximum and a local maximum. 8. What does the Extreme Value Theorem say? 9. What is the definition of a critical point? 10. What is the definition of an inflection point? 4

5 1. (a) 2 (b) 2/ (c) 1 (d) 1/5 (e) 0 (f) 0 (g) 1 2. The vertical asymptotes are x = 1 and x = 1. At x = 1, f approaches from the left and from the right. At x = 1, f approaches from the left and from the right.. The horizontal asymptotes are y = 2 and y = 2. As x, f(x) approaches 2, and as x, f(x) approaches Since 1 sin x 1, 1/x 2 (sin x)/x 2 1/x 2. Furthermore, 1 x x 2 = sin x by the squeeze theorem, x x 2 = a = 1, b = 6. (a) x = 5, 7, 9 (b) x =, 5, 7, 8, 9 7. f (x) = 2x cos(x 4 ) sin(x 4 ). x 1 = 0, so x2 8. sin(xy)(xy + y) = y cos(y). 9. y 2 = 7 2 (x 1). 10. Absolute minimum: -1, Absolute maximum: Local min at x = 1 and x = 1, local max at ( x = 0. f is ) increasing ( ) on ( 1, 0) (1, ), decreasing on (, 1) (0, 1), concave up on, 1 1,, and concave down ( ) on (a) f is positive on (2, 6) and (8, 10) since f is increasing on those intervals. (b) f is positive on (0, 5) and (7, 9) because f is concave up on those intervals. 1. F (x) = 2 sin x e x sec x 15. x 2 sin(x 6 ) Answers to Concept Questions 1. (a) f(x) + 2, (b) f(x + ), (c) f(x), (d) f(x/5), (e) f(2x) 2. arcsin x = y if and only if sin y = x. Its domain is 1 x 1 and its range is π/2 y π/2.. If g(x) f(x) h(x) near x = a and g(x) = h(x) = L, then f(x) = L. 4. f(x) = f(a). 5

6 5. f f(a + h) f(a) (a) = or f f(x) f(a) (a) =. The definition says that the slope h 0 h x a of the tangent line at x = a is the it of the slopes of the secant lines connecting (a, f(a)) to the points approaching it. 6. If the function is not continuous at a, if the function has a cusp/corner at a, or if the function has a vertical tangent line at a. (Another acceptable answer would be vertical asymptote, vertical tangent line, cusp, but I would not accept vertical asymptote, discontinuity, cusp or cusp, corner, discontinuity for example, since a vertical asymptote is a type of discontinuity, and a cusp/corner is referring to the same type of issue.) 7. An absolute maximum is the maximum y-value that a function achieves. A local maximum is a y-value on the function that is bigger than the ones near it (i.e. on an open interval around it). 8. The Extreme Value Theorem says that a function which is continuous on a closed interval achieves an absolute maximum and an absolute minimum on that interval. 9. A critical point of f is a value x such that f (x) = 0 or f (x) does not exist. 10. An inflection point of f is a point (x, f(x)) such that f changes concavity at x, i.e. f (x) changes sign. True-False problems: Chapter 2 Review: # 1,, 11, 19, 21 Chapter Review: # 1,, 5, 7, 15 Chapter 4 Review: # 1,, 5, 7, 11 6