12 Questions with Answer of Calculus and Analytic Geometry - Exam 1 | MATH 221, Exams of Analytical Geometry and Calculus

Download 12 Questions with Answer of Calculus and Analytic Geometry - Exam 1 | MATH 221 and more Exams Analytical Geometry and Calculus in PDF only on Docsity! 1 Practice Exam 1 A. Miller Spring 89 Math 221 limits 1. limx→∞ 1+x−2x 2 3x2−4x+1 2. limx→0 −3 sin2(2x) −x2 derivatives 3. Find y′′ if y = 5 12 x4 − 7 12 x3 + 1 2 x2 − 2x + 5 4. Show that if f(x) = √ x then f ′(x) = 1 2 √ x directly without using any rule for taking derivatives. 5. State the quotient rule for finding the derivative of u/v where u and v are functions of x. Give a proof of this formula directly from the definition of derivative and not using any other rule for taking derivatives. 6. Find dydx if y = (6− √ x)7/4. 7. Find dydx if y = tan(sin 2(x)). 8. Assume that y is a differentiable function of x. Use implicit differentiation to find dydx if 6x 3 + 8 cos(xy) = 2. 9. The following parametric equations give the position of a particle moving in the plane. y = t2, x = t3 − t Find d 2y dx2 at the point the particle is at when t = 1. tangent lines and linear approximation 10. Find the equation of the tangent line to the curve x2 +4xy− y2 = 11 at the point (2, 1). 11. Write an equation for the line tangent to y = x3 + 2x2 − 2 at the point (2, 14). 12. Let f(x) = −x3 − 3x + 2. Find the linearization of f at a = 1 use it to approximate f(1.01). 13. The curve y = ax2 + bx + c passes through the point (1, 4) and is tangent to the line y = 2x at the origin. Find a,b,and c. Newton’s method 14. Use Newton’s method to find x2 if f(x) = x3 +3x2 +4x+1 where x0 = −1. Practice Exam 1 A. Miller Spring 89 Math 221 2 True and False 15. (True or False) If f is a function which is continuous at a a then f is differentiable at a. 16. (True or False) If f is a function which is differentiable at every real x and f(a) < 0 and f(b) > 0, then there exist a c between a and b such that f(c) = 0. misc 17. Define f(−4) in a way that extends f(x) = x2 − 16 x2 − x− 20 to be continuous at x = −4. 18. Prove that (a)limx→0 sin(x) x = 1 or assume (a) and prove that (b) d dx sin(x) = cos(x). Exam from Feb 89 1. A particle is moving along a straight line at constant acceleration. At time t = 0 it has moved 0 units, at time t = 1 it has moved a total of 3 units, and at time t = 2 it has moved a total of 12 units. At time t = 3 what is its velocity? 2. For each of these say whether it is true or false. a. A continuous function on an interval is always differentiable except for at most finitely many points. b. If a function f(x) is differentiable at c then it is continuous at c. c. If f(x) = (x3 +x+1) then there must be a real number c with −1 < c < 1 such that f(c) = 0. d. The product of two differentiable functions is continuous. 3. Calculate the following limits or show they don’t exist: a. limx→0 sin(x) x b. limx→1 x−1x2−1 c. limx→0 x2cosec2(3x) d. limx→0 sin( 1x ) e. limx→∞ √ x2+1√ 2x2+x+1 4. a.Suppose y = 2x2 − 3x + 7. What is dydx? b. Suppose p = sin(y) sec(y). What is dpdy ? c. Suppose r = sin(2θ)tan(θ) . What is dr dθ? d. Suppose y = sin(x2 + x− 1). What is dydx? e. What is ddx ((x 2 + 1)5c) where c is a constant?