Differential Calculus application of calculus
- 1. Differential Calculus and its Application
- 2. Introduction 1. Derivative. The derivative of a function f at a number a is 𝑓′ 𝑎 = lim ℎ→0 𝑓 𝑎+ℎ −𝑓(𝑎) 𝑎 if this limit exists. 2. Tangent Line. An equation of the tangent line to 𝑦 = 𝑓(𝑥) 𝑎𝑡 (𝑎, 𝑓(𝑎)) is given by 𝑦 − 𝑓 𝑎 = 𝑓′ 𝑎 𝑥 − 𝑎 . 3. Product and Quotient Rules. If f and g are both differentiable, then 𝑓𝑔 ′ = 𝑓 ∙ 𝑔′ + 𝑓′ ∙ 𝑔 𝑎𝑛𝑑 𝑓 𝑔 ′ = 𝑔∙𝑓′−𝑓∙𝑔′ 𝑔2 , with 𝑔 ≠ 0. 4. Chain Rule. If f and g are both differentiable and 𝐹 = 𝑓 ∘ 𝑔 is the composite function defined by 𝐹(𝑥) = 𝑓(𝑔(𝑥)), then F is differentiable and F’ is given by 𝐹’(𝑥) = 𝑓′(𝑔(𝑥)) ∙ 𝑔′(𝑥). 5. Implicit Differentiation. Let a function y = y(x) be implicitly defined by F(x; y) = G(x; y). To find the derivative y0 do the following: (a) Use the chain rule to differentiate both sides of the given equation, thinking of x as the independent variable. (b) Solve the resulting equation for dy/dx.
- 3. 6. The Method of Related Rates. If two variables are related by an equation and both are functions of a third variable (such as time), we can and a relation between their rates of change. We say the rates are related, and we can compute one if we know the other. We proceed as follows: (a) Identify the independent variable on which the other quantities depend and assign it a symbol, such as t. Also, assign symbols to the variable quantities that depend on t. (b) Find an equation that relates the dependent variables. (c) Differentiate both sides of the equation with respect to t (using the chain rule if necessary). (d) Substitute the given information into the related rates equation and solve for the unknown rate.
- 6. Find the derivative of the following: 1. 𝑓 𝑥 = 𝑥2 sin2 (2𝑥2 ) 2. 𝑦 = sec 𝑥2 + 1 3. Find y’ when 𝑦 = 𝑒4𝑐𝑜𝑠ℎ 𝑥 4. Evaluate Dt cos−1 (cosh(𝑒−3𝑡 )), without simplifying your answer.
- 7. Related Rates A ladder 15 ft long rests against a vertical wall. Its top slides down the wall while its bottom moves away along the level ground at a speed of 2 ft/s. How fast is the angle between the top of the ladder and the wall changing when the angle is =3 radians? Ans: dƟ/dt = 4/15 ft /s
- 8. Related Rates A ladder 12 meters long leans against a wall. The foot of the ladder is pulled away from the wall at the rate ½ m/min. At what rate is the top of the ladder falling when the foot of the ladder is 4 meters from the wall? Ans: dy/dt = − 2 8 𝑚/𝑚𝑖𝑛
- 9. Related Rates • A rocket R is launched vertically and its tracked from a radar station S which is 4 miles away from the launch site at the same height above sea level. At a certain instant after launch, R is 5 miles away from S and the distance from R to S is increasing at a rate of 3600 miles per hour. Compute the vertical speed v of the rocket at this instant. Ans: v = dx/dt = 6000 mi/h
- 10. Related Rates • A boat is pulled into a dock by means of a rope attached to a pulley on the dock. The rope is attached to the bow of the boat at a point 1 m below the pulley. If the rope is pulled through the pulley at a rate of 1 m/sec, at what rate will the boat be approaching the dock when 10 m of rope is out.
- 11. Related Rates • An airplane is flying horizontally at an altitude of y = 3 km and at a speed of 480 km/h passes directly above an observer on the ground. How fast is the distance D from the observer to the airplane increasing 30 seconds later? Solution: D is the distance between observer and plane Based on Pythagorean Theorem: 𝐷2 = 32 + 480𝑡 2 𝐷2 = 9 + 230400𝑡2 2𝐷 𝑑𝐷 𝑑𝑡 = 460800𝑡 Since 30 𝑠 1𝑚𝑖𝑛 60𝑠 1ℎ 60 𝑚𝑖𝑛 = 1 120 ℎ𝑟 Substitute: 480 𝑘𝑚 ℎ𝑟 1 120 ℎ𝑟 = 4 𝑘𝑚 To solve for D, D = 32 + 42 = 5 𝑘𝑚 𝑑𝐷 𝑑𝑡 = 460800 1 120 hr 2 5 = 384 km/hr
- 12. Related Rates • A balloon is rising at a constant speed 4m/sec. A boy is cycling along a straight road at a speed of 8m/sec. When he passes under the balloon, it is 36 meters above him. How fast is the distance between the boy and balloon increasing 3 seconds later. Solution: Let x = distance between the boy and the balloon 𝑥2 = 8𝑡 2 + 36 + 4𝑡 2 2𝑥 𝑑𝑥 𝑑𝑡 = 128𝑡 + 2 36 + 4𝑡 4 = 128𝑡 + 8 36 + 4𝑡 𝑑𝑥 𝑑𝑡 = 128 3 + 8(36 + 4(3) 2(53.67) = 7.155 𝑚/𝑠 36 m 8 m/s 4 m/s x
- 13. Related Rates • Sand is pouring out of a tube at 1 cubic meter per second. It forms a pile which has the shape of a cone. The height of the cone is equal to the radius of the circle at its base. How fast is the sandpile rising when it is 2 meters high?
- 14. Related Rates • A water tank is in the shape of a cone with vertical axis and vertex downward. The tank has radius 3 m and is 5 m high. At rst the tank is full of water, but at time t = 0 (in seconds), a small hole at the vertex is opened and the water begins to drain. When the height of water in the tank has dropped to 3 m, the water is owing out at 2 m3/s. At what rate, in meters per second, is the water level dropping then?