Workbook for Calculus I

Transcription

1 Workbook for Calculus I By Hüseyin Yüce New York 2007

2 1 Functions 1.1 Four Ways to Represent a Function 1. Find the domain and range of the function f(x) = 1 + x + 1 and sketch its graph. y x Compute and then simplify the quantity f(a + h) f(a) for the function f(x) = x 2 2x. 3. Find the domain of the function f(x) = 1 x 2 3x 1

3 1.3 New Functions from Old Functions 1. For the functions f(x) = 1 x 1 their domain: 1 and f(x) =, Perform the following operations and find 2x + 1 (a) f + g (b) f g (c) f/g (d) f g 2. Write a function that is obtained by applying the following operations to f(x) = x: (i) shift 1 unit down, (ii) shift 2 units left, (iii) reflect about x-axis, and (iv) compress vertically by a factor of 3. 2

4 2 Limits and Rates of Change 2.2 The Limit of a Function 1. Which of the following statements about the function y = f(x) graphed below are true, and which are false? (a) lim f(x) = 1 x 1 + (b) lim x 2 f(x) does not exist. (c) lim x 2 f(x) = 2 (d) lim f(x) = 1 x 1 + (e) lim x 1 f(x) does not exist. (f) lim x 0 f(x) = 1 (g) lim x 0 f(x) does not exist. (h) f(0) = 1 2. Find the limit if it exits. If the limit does not exist, explain why: x 2 + x 2 (a) lim x 0 x 2 = 1 3 x (b) lim x 2 (x 2) 2 = 3

5 2.3 Calculating Limits Using the Limit Laws 1. Find the limit if it exits. If the limit does not exist, explain why: x 1 = x lim x 1 2. Find the limit if it exits. If the limit does not exist, explain why: x lim x 0 x = 3. Find the limit if it exits. If the limit does not exist, explain why: x + 3 lim x 2 x + 6 = 4

6 2.5 Continuity 1. Which of the following statements about the function y = f(x) graphed below are true, and which are false? (a) lim x 2 f(x) = 2 (b) lim x 1 f(x) does not exist. (c) f(x) is continuous at x = 2. (d) f(x) is continuous at x = 0. (e) f(x) is continuous from right at x = Show that the equation x 3 15x + 1 = 0 has at least one solution in the interval [ 4, 4] (hint: use the Intermediate Value Theorem). 3. For what value of a is f(x) = { x 2 1 if x < 3 2ax if x 3 continuous at every x? 5

7 2.6 Tangents, Velocities, and Other Rates of Change 1. Find an equation of the tangent line to the curve y = 2x x 2 at the point x = Find the slope of the tangent to the curve y = x 2 + x + 1 at the points x = a, x = 1, and x = 1 6

8 3 Derivatives 3.1 Derivatives 1. Estimate the slope of the curve (in y-units per x-unit) at the points P 1 and P If the tangent line to y = f(x) at (4,3) passes through the point (0,2), find f(4) and f (4). 3. Find f (a) for the function f(x) = x 2 5x. 7

9 3.2 The Derivative as a Function 1. The graph of y = f(x) is given below. Sketch a possible graph of its derivative f (x). 2. For the function f(x) = 1 x 2, find f (x) using the limit definition of derivative. 3. Find the derivative, f (x), of f(x) = 21 using the limit definition of derivative. 8

10 3.3 Differentiation Formulas 1. Differentiate f(x) = 5x + 1 4x 2 2. Differentiate f(x) = x3 2x + 5 x Differentiate f(x) = (2x 3 1)(7x 4 3x + 2) 9

11 3.5 Derivatives of Trigonometric Functions 1. Differentiate f(x) = sin x x Differentiate f(x) = x 2 cos x. 3. Find the following limit: tan 6x lim x 0 sin 3x = 10

12 3.6 The Chain Rule 1. Find an equation of the tangent line to the curve y = cos 2 x at the point x = π/4. 2. Find dy/dx for y = (2 sin 2 x) Differentiate f(x) = ( x 1 x) 3. 11

13 3.7 Implicit Differentiation 1. Find dy/dx, by implicit differentiation of, of y 2 (x y) = x 2 (x + y) 2. Find dy/dx, by implicit differentiation of, of x 4 + x 2 y 2 + y 4 = Find an equation of the tangent line to the curve xy = 8 at the point (4, 2). 12

14 3.8 Higher Derivatives 1. Find y of y = 2x + 1. x 2. Find f (x) of f(x) = x 2 sin x 3. Find f (x) of f(x) = sin 3x 13

15 3.9 Related Rates 1. A 13-ft ladder is leaning against a house when its base starts to slide away. By the time the base is 12 ft from the house, the base is moving at the rate of 5 ft sec. How fast is the top of the ladder sliding down the wall then? 2. When a circular plate of metal is heated in an oven, its radius increases at the rate of 0.01 cm/min. At what rate is the plates area increasing when the radius is 50 cm? 14

16 3.10 Linear Approximations and Differentials 1. Find the linearization L(x) of f(x) = x at the point a = Find the differential dy of y = x 2 + 3x and evaluate dy for the values of x = 3 and dx = Find the differential of the function y = 1 t 2. 15

17 4 Applications of Differentiations 4.1 Maximum and Minimum Values 1. Find the absolute extreme values of f(x) = 4 x 2 on the interval 3 x 1 2. Find the critical numbers of the function f(x) = x 4 + 4x What can you say about a function whose maximum and minimum values on an interval are equal? Give reasons for your answer. 16

18 4.2 The Mean Value Theorem 1. Show that the the given functions satisfy the the hypotheses of the mean value theorem on the indicated interval, and find all numbers c in that interval that satisfy the conclusion of that theorem. (a) f(x) = 3x 2 + 6x 5 on [ 2, 1] (b) f(x) = x + 1 x on [2, 3] 2. Show that the the function f(x) = x 4 + x 2 satisfy the the hypotheses of Rolle s theorem on the interval [0, 1], and find all numbers c in that interval that satisfy the conclusion of that theorem. 17

19 4.3 How Derivatives Affect the Shape of a Graph 1. Use the sign chart for the derivative df dx = 6(x 1)(x 2)2 (x 3) 3 to identify the points where f has local maximum and minimum values. 2. Sketch the graph of f(x) = 2x 3 3x 2 using the complete sign chart (calculator solutions will not receive any credit). 18

20 4.4 Limits at Infinity; Horizontal Asymptotes 4x 1. Find the limit lim 2 x x x Find all vertical and horizontal asymptotes of the function f(x) = 2x2 3 x 2 x y 2 3. Find the limit lim y 5y 2 + 4y. 19

21 4.5 Summary of Curve Sketching 1. Use the guidelines of curve sketching to sketch the curve f(x) = x x 2 1. y x

22 4.6 Graphing with Calculus and Calculators 1. Use calculus (with sign chart) to sketch the graph of f(x) = 8x 3 3x Use calculator to sketch the graph of f(x) = x2 + 11x 20 x 2. 21

23 4.7 Optimization Problems 1. An open rectangular box is made from a square piece of material 6 inches on a side, by cutting equal squares from each corner and turning up the sides. Find the dimensions of the box that will give the maximum volume. 2. What is the largest possible area for a right triangle whose hypotenuse is 5 cm long? 22

24 4.10 Antiderivatives 1. If f (x) = 8x x + 3 and f(1) = 6, find f(x). 2. Find the most general antiderivative of the function f(x) = 24x 2 + 2x If f (x) = 2 + cos x, then find f(x). 23

25 5 Integrals 5.1 Areas and Distances 1. Estimate the area under the graph of f(x) = 1 + x from x = 0 to x = 4 with n = 4 rectangles using LHS and RHS. 2. The marginal cost C (q) (in dollars per unit) of producing q units is given in the following table. If fixed cost is $5000, estimate the total cost of producing 200 units. q C (q)

26 5.3 The Fundamental Theorem of Calculus 1. Find the derivative of G(x) = x t 3 dt 2. Evaluate the integral 9 4 x dx 3. The graph of y = f (x) (NOT f(x)) is given below. If f(0) = 5, using Fundamental Theorem of Calculus, (a) Find f(1). (b) Find f(4). (c) Find f(6). (d) Find 5 0 f (x) dx. 25

27 5.5 The Substitution Rule 1. Evaluate the following integrals: (a) π/2 0 cos 2x dx (b) x 3 x dx π 0 sin 2 x cos x dx 26

28 6 Applications of Integration 6.1 Areas Between Curves 1. Find the area between the curves y = x 2 2x and y = x Determine the area of the region bounded by the curve y = x x 24 and the x-axis. 27