SMA 2101: CALCULUS

SMA 2101: CALCULUS I ⃝c Francis O. Ochieng Department of Pure and Applied Mathematics Jomo Kenyatta University of Agriculture and Technology Course content • Trigonometric functions: addition, multiple angle and factor formulae. • Limits, continuity and differentiability. • Differentiation by first principles and by rule for x n (integral and fractional n), sums, products, quotients, chain rule, trigonometric, logarithmic and exponential functions of a single variable. Parametric differentiation. • Applications: equations of tangent and normal, kinematics, rates of change and stationary points, economics and financial models. • Integration: anti-derivatives and their applications to areas, volumes, and economics and financial models. References [1] Calculus and Analytical Geometry by Thomas and Funnery [2] Advanced Engineering Mathematics (10th ed.) by Erwin Kreyszig [3] Calculus by Larson Hostellem Lecture 1 1 Introduction Calculus is a Latin name meaning stone used for calculation. We therefore deal with calculus as infinitesimal calculus meaning calculation of numbers which are infinitesimally small. For example, consider the growth of a sidling - it grows gradually and continuously. If observed after few days, the growth can be measured, but if observed after few minutes, although growth has taken place the amount is too small to be distinguished. Therefore, it has an infinitesimal growth. In calculus, the amount of growth is not important but the rate of growth is important. 1.1 Functions The growth of a sidling is an instance of a functional relation, in that the growth may be affected by variations in temperature, moisture, sunlight, etc. If all these factors remain constant, then the growth is a function of time. To understand the word function, we first look the following definitions. Definition 1.1 (Variables and constants). Consider the formula used for calculating the volume of a sphere of radius r. V = 4 3 πr3 (1) Then, 1 Downloaded by Brian Muchoki () lOMoARcPSD| i) V and r vary with different spheres. Hence, they are called variables. ii) π and 4 3 are constants, irrespective of the size of the sphere. Definition 1.2 (Dependent and independent variables). From the formula (1), the volume V depends on value of the radius, r, of the sphere. In this case, r is called the independent variable (input) since it is not affected while V is called the dependent variable (output) since it is affected by the variation of r. Similarly, in a second degree equation y = ax2 + bx + c. Here, a, b, c are constants, x is the independent variable, and y is the dependent variable. Definition 1.3 (Calculus). This is the study of the effect of change in the input (independent variable) on the output (dependent variable). Definition 1.4 (Function). This is a mathematical relation between the dependent and independent variable. For example, i) If the variable y dependents and the variable x e.g., y = 2x 2 − 3x + 5, then we say that dependent variable y is a function of the independent variable x and we denote y = f(x), where f is a function and x is the independent variable. We may also denote this functional relation as y = ϕ(x) or y = ψ(x) or y = g(x), etc. ii) From formula (1), the volume V of a sphere is a function of the radius r - i.e., V = f(r). iii) In the formula S = 1 2 gt2 , the distance S moved by a falling body is a function of time t. iv) The volume of a fixed mass of a gas is a function of the absolute temperature while the pressure remains constant, and vise versa. v) The sine, cosine, tangent of an angle are functions of the angle. Example(s): (a) If a function y = f(x) = x 2 − 4x + 3. Find (i) f(1), (ii) f(2), (iii) f(a), and (iv) f(a + h). Solution i) f(x) = x 2 − 4x + 3 ⇒ f(1) = 12 − 4(1) + 3 = 0 ii) f(x) = x 2 − 4x + 3 ⇒ f(2) = 22 − 4(2) + 3 = −1 iii) f(x) = x 2 − 4x + 3 ⇒ f(a) = a 2 − 4a + 3 iv) f(x) = x 2 − 4x + 3 ⇒ f(a + h) = (a + h) 2 − 4(a + h) + 3 (b) If a function ϕ(θ) = 2 sin θ, find (i) ϕ( π 2 ), (ii) ϕ(0), and (iii) ϕ( π 3 ). Solution i) ϕ(θ) = 2 sin θ ⇒ ϕ( π 2 ) = 2 sin ( π 2 ) = 2 ii) ϕ(θ) = 2 sin θ ⇒ ϕ(0) = 2 sin (0) = 0 iii) ϕ(θ) = 2 sin θ ⇒ ϕ( π 3 ) = 2 sin ( π 3 ) = 2 × √ 3 2 = √ 3 Exercise: (a) If f(x) = 2x 2 − 4x + 1, find (i) f(1), (ii) f(0), (iii) f(2), (iv) f(a), and f(x + h). (b) If f(x) = (x − 1)(x + 5), find (i) f(1), (ii) f(0), (iii) f(2), (iv) f(a + 1), and f( 1 a ). (c) If f(θ) = cos θ, find (i) f( π 2 ), (ii) f(0), (iii) f( π 3 ), (iv) f( π 6 ), and (v) f(π). (d) If f(x) = x 2 , find (i) f(3), (ii) f(3.1), (iii) f(3.01), (iv) f(3.001), and f(3.001) − f(3) 0.001 . (e) If ϕ(x) = 2x , find (i) ϕ(0), (ii) ϕ(1), and (iii) ϕ(0.5). 2 Downloaded by Brian Muchoki () lOMoARcPSD| 1.2 Limits of functions Definition 1.5 (Basic limit definition). Let f(x) be a function and let a and L be real numbers. If f(x) approaches L as x approaches a (but is not equal to a), then we say that f(x) has limit L as x approaches a and we denote lim . x→a f(x) = L. → Note: limx→a f(x) is the value that f(x) approaches as x approaches a, and a does not have to be in the domain of f. 1.3 Properties of limits Theorem 1.1. Suppose limx→a f(x) = L1 and limx→a g(x) = L2, then 1. [Addition rule] limx→a [f(x) + g(x)] = L1 + L2 2. [Scalar multiple] limx→a [λf(x)] = λL1, where λ is a constant. 3. [Product rule] limx→a [f(x) · g(x)] = L1 · L2 4. [Quotient rule] limx→a [ f(x) g(x) ] = L1 L2 1.4 Methods of evaluating limits of functions  Direct substitution The required limit is obtained by just plugging in the value of input, say x, into the given function, say f(x). Example(s): (a) Evaluate limx→2 3x 3 − x 2 + 2x + 5. Solution limx→2 (3x 3 − x 2 + 2x + 5) = 3 limx→2 x 3 − limx→2 x 2 + 2 limx→2 x + limx→2 5) = 3(23 ) − (22 ) + 2(2) + 5 = 29 (b) Evaluate limx→1 x 2 − 1 x + 1 . Solution limx→1 x 2 − 1 x + 1 D.S= 1 2 − 1 1 + 1 = 0 2 = 0  Factorization If on direct substitution we get the indeterminate form 0/0, then it means that there is a common factor in both the numerator and denominator. In this case, we perform factorization first so as to simplify the given function. → Note: if the polynomial in the numerator is of degree greater than the degree of the polynomial in the denominator, we first need to perform long division. Example(s): 3 Downloaded by Brian Muchoki (