Math 1131Q Section 10 Section 3.9 and 3.10 Oct 19, 2010
Find the derivative of ln 3 5 e 2 ln 3 5 e 2 = ln 3 + ln 5/2 + ln e 2 = 3 ln + ( 5 ) ln + 2 2 (ln 3 5 e 2 ) = 3 + 5 2 + 2 Find the derivative of t 3 2t + 5 t t 3 2t + 5 t = t3 2t t1/2 t + 5 1/2 t = 1/2 t5/2 2t 1/2 + 5t 1/2 ( t3 2t + 5 ) = ( 5 t 2 )t3/2 t 1/2 5 2 t 3/2 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 2 / 25
Clicker Question Find dy d given 3 3y 2 + y 3 = 5 (a) y = 32 + 3y 2 (3y 2 6y) (b) y = 32 + 3y 2 (3y 2 6y) (c) y = 32 3y 2 (3y 2 + 6y) (d) y = 32 + 3y 2 (3y 2 + 6y) Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 3 / 25
3 3y 2 + y 3 = 5 3 2 3(y 2 + 2yy ) + 3y 2 y = 0 3 2 3y 2 6yy ) + 3y 2 y = 0 6yy + 3y 2 y = 3 2 + 3y 2 (3y 2 6y)y = 3 2 + 3y 2 y = 32 + 3y 2 (3y 2 6y) Answer = a Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 4 / 25
d d (sin 1 ) = Two views 1 1 2 y = sin 1 = sin y d d () = d (sin y) d 1 = cos y dy d dy d = 1 cos y Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 5 / 25
d d (sin 1 ) = Two views 1 1 2 y = sin 1 = sin y d d () = d (sin y) d 1 = cos y dy d dy d = 1 cos y sin (arcsin ) = Take the derivative of both sides: cos (arcsin ) d (arcsin ) = 1 d Solve for d (arcsin ) d d d (arcsin ) = 1 cos (arcsin ) Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 5 / 25
dy d = 1 cos y d d (arcsin ) = 1 cos (arcsin ) θ 1 1 2 y = sin 1 is the angle with cos y = cos (sin 1 ) = 1 2 dy d = 1 cos y d d (arcsin ) = 1 cos (arcsin ) dy d = d d (arcsin ) 1 1 2 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 6 / 25
Derivatives of f 1 () The general form y = f 1 () = f(y) d d () = d d (f(y)) 1 = f (y) dy d dy d = 1 f (y) where y = f 1 () d d (f 1 ()) = 1 f (f 1 ()) Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 7 / 25
Derivatives of f 1 () The general form y = f 1 () = f(y) d d () = d d (f(y)) 1 = f (y) dy d If f(a) = b then f 1 (b) = a dy d = 1 f (y) where y = f 1 () d d (f 1 ()) = 1 f (f 1 ()) and d d (f 1 (b)) = 1 f (f 1 (b)) = 1 f (a) Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 7 / 25
d d (f 1 ()) = 1 f (f 1 ()) If f() = find the slope of the tangent line to the graph of f 1 at the point (2, 4). Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 8 / 25
d d (f 1 ()) = 1 f (f 1 ()) If f() = find the slope of the tangent line to the graph of f 1 at the point (2, 4). d d (f 1 (2)) = f () = 1 2 1 f (f 1 (2)) = 1 f (4) Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 8 / 25
d d (f 1 ()) = 1 f (f 1 ()) If f() = find the slope of the tangent line to the graph of f 1 at the point (2, 4). d d (f 1 (2)) = f () = 1 2 d d (f 1 (2)) = 1 1 1 f (f 1 (2)) = 1 f (4) 2 4 = 2 4 = 4 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 8 / 25
Clicker Question A boat is drawn close to a dock by pulling in the rope at a constant rate. The closer the boat gets to the dock, the faster it is moving. (a) TRUE (b) FALSE (c) Not enough information Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 9 / 25
Clicker Question A boat is drawn close to a dock by pulling in the rope at a constant rate. The closer the boat gets to the dock, the faster it is moving. (a) TRUE (b) FALSE (c) Not enough information a) True - The closer the boat gets to the dock, the faster it is moving. Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 9 / 25
Clicker Question A boat is drawn close to a dock by pulling in a rope as shown. How is the rate at which the rope is pulled in related to the rate at which the boat approaches the dock? (a) One is a constant multiple of the other. (b) They are equal. (c) It depends on how close the boat is to the dock. h z dz d h is constant Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 10 / 25
Clicker Question A boat is drawn close to a dock by pulling in a rope as shown. How is the rate at which the rope is pulled in related to the rate at which the boat approaches the dock? (a) One is a constant multiple of the other. (b) They are equal. (c) It depends on how close the boat is to the dock. h (c) The relationship between d boat is to the dock. z and dz dz d h is constant depends on how close the Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 10 / 25
Let h be the height of the dock above the point on the boat to which the rope is tied. h z Then, using the Pythagorean Theorem, h 2 + ((t)) 2 = (z(t)) 2 and differentiating, one gets 2(t) d = 2z(t)dz d = z(t) dz (t) dz = constant As (t) 0 we have z(t) h. Hence z(t) (t) d and Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 11 / 25
A streetlight is mounted at the top of a 12 ft pole. A man 5 ft tall walks away from the pole with a speed of 4 ft/sec. How fast is the tip of his shadow moving when he is 24 feet from the pole? y Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 12 / 25
A streetlight is mounted at the top of a 12 ft pole. A man 5 ft tall walks away from the pole with a speed of 4 ft/sec. How fast is the tip of his shadow moving when he is 24 feet from the pole? y We are given that the rate of change of y is 4 ft/sec and we want to find the rate of change of z = + y. Notice that d is how fast the length of the shadow is increasing, which is slower by the speed of the man walking. Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 12 / 25
We first draw a figure and determine the relationship between, y and z. We are given dy dz = 4 ft/sec and want to find. y 7 12 y z 5 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 13 / 25
We first draw a figure and determine the relationship between, y and z. We are given dy dz = 4 ft/sec and want to find. y 7 12 y z 5 z 12 = y 7 z = 12 7 y Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 13 / 25
We first draw a figure and determine the relationship between, y and z. We are given dy dz = 4 ft/sec and want to find. y 7 12 y z 5 z 12 = y 7 z = 12 7 y Take the derivative of both sides dz = 12 dy 17 dz = 12 48 (4) = 7 7 ft/sec Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 13 / 25
Clicker Question A streetlight is mounted at the top of a 12 ft pole. A man 5 ft tall walks away from the pole with a speed of 4 ft/sec. How fast is the length of his shadow increasing when he is 24 feet from the pole? (a) 48/7 (b) 20/7 (c) 68/7 (d) 4 (e) None of these y 7 12 y z 5 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 14 / 25
Clicker Question A streetlight is mounted at the top of a 12 ft pole. A man 5 ft tall walks away from the pole with a speed of 4 ft/sec. How fast is the length of his shadow increasing when he is 24 feet from the pole? (a) 48/7 (b) 20/7 (c) 68/7 (d) 4 (e) None of these y 7 12 y z 5 b) d = dz dy d since = z y so = 48/7 4 = 20/7 Or 5 = y = = 5 7 7 y = d = 5 dy 7 = 20 7 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 14 / 25
Related rate problems Draw a picture and label variables h z Epress the relationship between the variables. h 2 + ((t)) 2 = (z(t)) 2 Take the derivative with respect to t to find the relationship between the derivatives. 2(t) d = 2z(t) dz Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 15 / 25
Clicker Question The volume of a cube is increasing at the rate of 10 cubic cm per min. How fast is the surface area increasing when the length of an edge is 30 cm? (a) 1/3 (b) 2/3 (c) 4/3 (d) 5/3 (e) None of these Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 16 / 25
Clicker Question The volume of a cube is increasing at the rate of 10 cubic cm per min. How fast is the surface area increasing when the length of an edge is 30 cm? (a) 1/3 (b) 2/3 (c) 4/3 (d) 5/3 (e) None of these dv ds = 10 cu cm/min =? when = 30 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 16 / 25
S = 6 2 V = 3 ds = 12d dv When = 30 = 32 d 10 = 3(30) 2 d d = 10 3(30) 2 When = 30 ds = 12(30) 10 3(30) 2 ds = 4 3 cm3 /min Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 17 / 25
Clicker Question Two cars start moving from the same point. One travels south at 28 mph and the other one west at 70 mph. At what rate is the distance between the two cars increasing two hours later? Round the result to the nearest hundreth. (a) 75.42 mph (b) 75.49 mph (c) 75.38 mph (d) 76.4 mph (e) 75.39 mph Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 18 / 25
Clicker Question Two cars start moving from the same point. One travels south at 28 mph and the other one west at 70 mph. At what rate is the distance between the two cars increasing two hours later? Round the result to the nearest hundreth. Steps (a) 75.42 mph (b) 75.49 mph (c) 75.38 mph (d) 76.4 mph (e) 75.39 mph 1 Draw a figure and label 2 Find epressions relating variables 3 Differentiate with respect to t to find the relationship among the rates of change 4 Solve for the desired rate of change Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 18 / 25
Step 1. z y Step 2. 2 + y 2 = z 2 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 19 / 25
Step 1. z y Step 2. Step 3. 2 + y 2 = z 2 d (2 + y 2 ) = d (z2 ) 2 d + 2y dy = 2z dz Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 19 / 25
Step 1. z Step 2. Step 3. y 2 + y 2 = z 2 d (2 + y 2 ) = d (z2 ) 2 d + 2y dy = 2z dz Step 4. d dz d dz = + y dy z = 70 and dy = 28 = 140(70) + 28(56) 22, 736 dz 75.39 mph Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 19 / 25
A plane is flying horizontally at an altitude of 4 miles and at a speed of 465 miles per hour as it passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 10 miles away from the station Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 20 / 25
A plane is flying horizontally at an altitude of 4 miles and at a speed of 465 miles per hour as it passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 10 miles away from the station Step 1. y 000 111 000 111 4 Step 2. 2 + 16 = y 2 d = 465 mph Find dy when y = 10 miles. Step 3. Differentiate Step 4. Solve for dy when y = 10 miles Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 20 / 25
Clicker Question A plane is flying horizontally at an altitude of 4 miles and at a speed of 465 miles per hour as it passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 10 miles away from the station (a) About 426 mph (b) About 436 mph (c) About 406 mph (d) More than 427 mph (e) Less than 405 mph y 000 111 000 111 4 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 21 / 25
Clicker Question A plane is flying horizontally at an altitude of 4 miles and at a speed of 465 miles per hour as it passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 10 miles away from the station (a) About 426 mph (b) About 436 mph (c) About 406 mph (d) More than 427 mph (e) Less than 405 mph Answer = a, about 426.2 mph y 000 111 000 111 4 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 21 / 25
Step 1. 000 111 000 111 y d = 465 mph Find dy when y = 10 miles. Step 2. 2 + 16 = y 2 Step 3. Differentiate 2 d 4 = 2y dy 2 d = 2y dy Step 4. Solve for dy when y = 10 miles 2 + 16 = (10) 2 = = 84 d = 465 dy = 84 465 426.2 10 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 22 / 25
A water trough is 100 cm long and a cross-section has the shape of an isosceles triangle that is 75 cm wide at the top, and has a height of 55 cm. Find the volume of water in the trough when the depth of the water is h? Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 23 / 25
A water trough is 100 cm long and a cross-section has the shape of an isosceles triangle that is 75 cm wide at the top, and has a height of 55 cm. Find the volume of water in the trough when the depth of the water is h? 75 b h 55 Area of end = hb/2 b h = 75 similar triangles 55 A = 1 2 hb = 1 2 (15 11 )h2 Volume = area of end 100 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 23 / 25
A water trough is 100 cm long and a cross-section has the shape of an isosceles triangle that is 75 cm wide at the top, and has a height of 55 cm. Find the volume of water in the trough when the depth of the water is h? 75 b h 55 Area of end = hb/2 b h = 75 similar triangles 55 A = 1 2 hb = 1 2 (15 11 )h2 Volume = area of end 100 V = ( 15 22 )h2 (100) V = ( 1500 22 )h2 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 23 / 25
Clicker Question A water trough is 100 cm long and a cross-section has the shape of an isosceles triangle that is 75 cm wide at the top, and has a height of 55 cm. If the trough is being filled with water at the rate of 750 cm 3 per min, how fast is the water level rising when the water is 45 cm deep? Round the result to the nearest (a) 1.03 cm 3 per min (b) 10 cm 3 per min (c) 0.27 cm 3 per min (d) 0.122 cm 3 per min (e) None of these Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 24 / 25
V = 1500 22 h2 Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 25 / 25
V = 1500 22 h2 dv = 21500 22 hdh 750 = 2 1500 22 (45)dh dh = 750 2( 1500)(45) 22 dh = 1.22 cm3 /min Answer = d Section 3.9 and 3.10 (Oct 19, 2010) Math 1131Q Section 10 25 / 25