Chapter 1 Review of Straight Lines
1.1
Find the slope and y intercept of the following straight lines: (a) y = 4x 5 (b) 3x 4y = 8 (c) 2x = 3y (d) y = 3 (e) 5x 2y = 23
1.2
Find the equations of the following straight lines (a) Through the points (2,0) and
Chapter 12 Approximation methods
12.1 An approximation for the square root
Use a linear approximation to nd a rough estimate of the following functions at the indicated points. (a) y = x at x = 10. (Use the fact that 9 = 3.)
(b) y = 5x 2 at x = 1. (c) y =
Chapter 11 Inverse Trigonometric functions
11.1
The function y = arcsin(ax) is a so-called inverse trigonometric function. It expresses the same relationship as does the equation ax = sin(y ). (However, this function is dened only for values of x between
Chapter 10 Trigonometric functions
10.1
Calculate the rst derivative for the following functions. (a) y = sin x2 (b) y = sin2 x (c) y = cot2 3 x (d) y = sec(x 3x2 ) (e) y = 2x3 tan x (f) y =
x cos x
(g) y = x cos x (h) y = e sin
21 x
(i) y = (2 tan 3x + 3
Chapter 9 Exponential Growth and Decay: Dierential Equations
9.1
A dierential equation is an equation in which some function is related to its own derivative(s). For each of the following functions, calculate the appropriate derivative, and show that the
Chapter 8 Exponential functions
8.1
Graph the following functions: (a) f (x) = x2 ex (b) f (x) = ln(x2 + 3) (c) f (x) = ln(e2x )
8.2
Express the following in terms of base e: (a) y = 3x (b) y =
1 7x
2 +2
(c) y = 15x
Express the following in terms of base
Chapter 7 The Chain Rule, Related Rates, and Implicit Dierentiation
7.1
For each of the following, nd the derivative of y with respect to x. (a) y 6 + 3y 2x 7x3 = 0 (b) ey + 2xy = 3 (c) y = xcos x
7.2
Consider the growth of a cell, assumed spherical in sh
Chapter 6 Optimization
6.1
The sum of two positive number is 20. Find the numbers (a) if their product is a maximum. (b) if the sum of their squares is a minimum. (c) if the product of the square of one and the cube of the other is a maximum.
6.2
A tram r
Chapter 5 What the Derivative tells us about a function
5.1
A zero of a function is a place where f (x) = 0. (a) Find the zeros, local maxima, and minima of the polynomial y = f (x) = x3 3x (b) Find the local minima and maxima of the polynomial y = f (x)
Chapter 4 The Derivative
4.1
You are given the following information about the signs of the derivative of a function, f (x). Use this information to sketch a (very rough) graph of the function for 3 < x < 3. x f (x) -3 -2 -1 0 1 0+0-0 23 +
4.2
You are giv
Chapter 3 Average velocity, Average Rates of Change, and Secant Lines
3.1
A certain function takes values given in the table below. t 0 0.5 1.0 1.5 2.0 f (t) 0 1 0 -1 0 Find the average rate of change of the function over the intervals (a) 0 < t < 0.5, (b
Chapter 2 Review of Simple Functions
2.1
(a) On the same set of axes, sketch the functions y = x, y = x2 , and y = x3 for values of x greater than 0. Pay particular attention to the shapes of these graphs in the ranges [0,1] on the x axis and for x > 1. W
Chapter 13 More Dierential Equations
13.1
Consider the dierential equation dy = a by dt where a, b are constants. (a) Show that the function a Cebt b satises the above dierential equation for any constant C . y (t) = C= we also satisfy the initial conditi