Lecture-4 (1) - MATH 401 DIFFERENTIAL CALCULS FINALS Application of the Derivatives of Transcendental Functions Partial Derivatives Vector Analysis


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MATH 401: DIFFERENTIAL CALCULS FINALS: Application of the Derivatives of Transcendental Functions, Partial Derivatives, Vector Analysis and Introduction to Differential Equations, Solutions of Equations, Parametric Equations Application of the Derivatives of Transcendental Functions A. Exponential Growth/Decay If a positive function f(t) represents an amount of substance present at time t, then: f’(t) = kf(t) where k is a constant and t is greater than or equal to 0. Applying the integration, this equation further translates to ln[f(t)] = kt + c. Taking the antilogarithm, f(t) = Ae kt where A = e c is a positive constant If f(t) increases, then the function is an exponential law of growth. But if f(t) decreases, in cases where k is negative, then the function is an exponential law of decay. Examples: 1. Determine the amount of radium left after 20 years of decaying if it starts with 100 grams and is reduced to 85 grams after 13 years. Ans: 77.8778 g 2. If a bacterial colony starts with 1 x 10 3 as counted from a laboratory equipment, determine how much bacteria have multiplied based on starting population for 5 minutes, if after 3 minutes, the population becomes 3 x 10 8 . Ans: 1.3444 x 10 9 3. A frog laid eggs on the stagnant water fountain near CEAFA Building. The number of tadpoles hatched varies exponentially with time. If initially at 7:00 AM, there is a single tadpole, what time shall a Spartan visit this fountain to assure that the fountain is completely proliferated with tadpoles? Note that it takes 30 minutes for the tadpoles to completely fill half of the volume of the water. This equates to 87 tadpoles. A completely filled fountain will have 174 tadpoles. Ans: 7:34:39 AM B. Optimization Transcendental functions can also be applied for optimization problems including those which use trigonometric and inverse trigonometric functions. Examples: 1. Find the volume of the largest right circular cone which can be inscribed in a sphere of radius 9 inches. Let θ be the angle sub tending the radius of the base of the cone. Ans: 904.7720 in 3 2. Find the area of the largest rectangle that has one side on the x-axis and two vertices on the curve ? = ? −? 2 Ans: 1.1658 units 2 3. A statue 10 feet high is standing on a base 13 feet high. If an observer’s eye is 5 feet above the ground, how far should he stand from the base in order that the angle between his lines of sight to the top and bottom of the statue be a maximum? Ans: 12 ft 4. The range of projectile up an inclined plane is given by 𝑅 = 2? 2 ???𝜃sin (𝜃 − 𝛼) ???? 2 𝛼 where v and g are constants and α is the inclination of the plane with the horizontal. Find the angle of projection θ which makes R a maximum. Ans : π/4 + α/2 5. Prove that the curve y = 2cosx passes through all the points of inflection of the curve y = xsinx. Ans: The POI (10.6429, 1.9656) lies on the curve