University:
North Central College
Course:
MATH 152 | Calculus II
Academic year:

2024
Views:

362

Pages:

3
Author:

nesikhiyawl0

Question 1 Given the cone z = √ x² + y² and the plane z=5+y. Represent the curve of intersection of the surfaces with a vector function r(t). Answer x = f(t), y = g(t), and z = h(t) The vector: r(t) = f(t)i+g(t)j+h(t)k Consider the equation of cone surface: z = √x2 + y2 (a) Substitute z=5+y to (a) 5 + y = √x²+ y² (5 + y)² = (√x2+y2)2 25 + 10y + y² = x² + y² 25 + 10y = x² 10y = x² - 25 1 y= (x²-25) 10 y= 1 10 x² - 5 2 Consider the equation for plane surface: z=5+y (b) 1 5 Substitute y = 2 x to (b) 10 2 z=5+ 1 10 x² - 5 2 z= 1 10 5 2 x²+ To define each variable in terms of the parameter t, set x=t y= 1 10 t² - 5 2 1 and y = 10 t² + 5 2 Thus, r(t) = (t)i+( 1 10 t² - 5 2 )j+( 1 10 t² + 5 2 )k Question 2 Simplify tan (sin-1(x)) Answer 1 Let sin¯¹(x) = 0, then x = sin 0 If we draw a right triangle with hypotenuse equal to 1 and the other side equals to x. From pythagorean theorem: √ 1 - x2 is the other side. Thus, tan 0 = sin 0 / cos 0 = sin 0 / √1- sin20 As x = sin 0 Now, we have: tan 0 = x / √1-x² From sin¯¹(x) = 0 we get tan (sin¯¹(x)) = x / √1-x² Answer 2 tan (arcsin (x)) = x√1-x² / 1-x² Question 3 Find the exact value of each expression. (tan) 17π / 12 Answer Split up the argument of the trigonometric function and then use the Sum formula for the Tangent function, tan (u + v) = tan (u) + tan (v) / 1 - tan (u)tan (v) , to find the value of given expression. tan (17π / 12) = tan (9π / 12 + 8π / 12) = tan (3π / 4 + 2π / 3) = tan (tan(3π / 4) + tan(2π / 3) / 1-(tan(3π / 4)tan(2π / 3) ) = -1+(-√3) / 1-(-1)-(-√3) = -1-√3 / 1-√3 = (-1-√3)(1+√3) / (1-√3)(1+√3) = -1-2√3-3 / 1-3 = -4-2√3 / -2 = 2 + √3 Result: tan (17π / 12) = 2 + √3 Question 4 Which pair of angles has congruent values for the sin xº and the cos y°? α. 35°; 55° b. 35°; 145° c. 35°; 70° d. 35°; 35° Answer Step 1 Note: cos (900-θ) = cos (90) cos θ + sin (90) sin θ = 0 × cos θ + 1sin θ cos (900-θ) = sin θ Step 2 Using, it θ = 35, then cos (90-35) = sin (35) ∴ cos (55) = sin (35) ∴ 35°, 55° ∴ option a is correct. Question 5 What is the antiderivative of sec (x)? Answer ∫ sec xdx Multiply by sec (x) + tan(x) / sec (x) + tan(x) And we have: = ∫ sec (x)(sec (x) + tan(x)) / sec (x) + tan(x) dx = ∫ sec²(x) + sec (x)tan(x) / sec (x) + tan(x) dx = ∫ 1 / u du = ln|u|+C = ln|sec (x) + tan(x)| + C