MIDTERM EXAM 1 Math 19A - Mathematics 1-2-3

Math 19A MIDTERM EXAM 1 Math 19A10/21/2011 , Dr. Frank Bäuerle, UCSCNotes: Show your work. In other words, just writing the answer,even if correct, may not be sufficient for full credit. Scientific calculatorsare allowed, but no programmable and/or graphing calculators.And please put away your cell phones and other electronic devices,turned off or in airplane mode.Your Name:Your TA’s Name:Problem 1: out of 20Problem 2: out of 10Problem 3: out of 15Problem 4: out of 10Problem 5: out of 10Problem 6: out of 10Problem 7: out of 10Problem 8: out of 15Total: out of 100Good luck and have a relaxed weekend!1

1. (20 points) Compute the requested derivatives of the functions. NONEED TO SIMPLIFY.(a) y ′ (x) for y(x) = 4x 3 − 2x + 5(b) f ′ (t) for f(t) =3√t + 12 − t(c) dydxfor y = sin x cot x sec x(d) dfdz for f(z) = 9√ z(sin z + cos z)2

2. (10 points) An object with (constant) weight W is dragged along ahorizontal plane by a force F acting along a rope attached to the object.If the rope makes an angle θ with the plane, then the magnitude of theforce is given byF =µWµ sin θ + cos θwhere µ is a positive constant called the coefficient of friction.(a) Find the rate of change of F with respect to θ.(b) For what angle θ is this rate of change equal to 0?(c) (Extra Credit, 2 points) What is the significance of the value youfound in part (b)?3

4. (10 points) Let a, b be real numbers. Find the parabola with equationy = ax 2 + bx whose tangent line at the point (2, 14) has slope 15.5. (10 points) Use the Intermediate Value Theorem to show that the equationcos πx = x 2has at least two real solutions.(Extra Credit, 2 points: How many real solutions does the equationhave?)5

6. (10 points) Compute the equation of the tangent line to the curvey = 7e x cos xwhen x = 0. Give your answer in the form y = mx + b.7. (10 points) At what point(s) on the curve y = x 3 − 6x 2 + x + 5 is thetangent line parallel to the line with equation x − y = 7?6

8. (15 points) Use the appropriate limit laws and knowledge about specialsin xlimits such as lim = 1 to find the following limits. Justify yourx−→0 xanswer and if a limit does not exist, explain why.(a) (3 points)limx−→∞3x 2 + 12x 3 + 5x − 1(b) (4 points)lim x sin xx−→∞(c) (4 points)√lim 4x2 + x − 2xx−→∞(d) (4 points) limh−→0cos h − 1h7