CLASS QUIZ: OCTOBER 8: DERIVATIVESMATH 152, SECTION 55 (VIPUL NAIK)Your name (print clearly in capital letters):Write your answer in the space provided. In the space below, you can explain your work if you want (thiswill not affect scoring). I may or may not get time to look at the work you have done, but it may help yourecall how you arrived at a particular answer.You are expected to take about one minute per question.(1) Consider the expression x2+t2+xt. What is the derivative of this with respect to x (with t assumedto be a constant)?(A) 2x + 2t + x + t(B) 2x + 2t + 1(C) 2x + 2t(D) 2x + t + 1(E) 2x + tYour answer:(2) For a function f : (0, ∞) → (0, ∞), denote by f(k)the kthderivative of f . Supp ose f (x) := xrwith domain (0, ∞), and r a rational number. What is the precise set of values of r satisfyingthe following: there exist a positive integer k (dependent on r) for which f(k)is identically the zerofunction.(A) r should be an integer.(B) r should be a nonnegative integer.(C) r should be a positive integer.(D) r should be a nonnegative rational number.(E) r should be a positive rational number.Your answer:PLEASE TURN OVER FOR THE THIRD AND FOURTH QUESTION.1(3) Which of the following s tateme nts is definitely true about the tangent line to the graph of aneverywhere differentiable function f on R at the point (a, f(a)) (Here, “everywhere differentiable”means that the derivative of f is defined and finite for all x ∈ R)?(A) The tangent line intersects the curve at precisely one point, namely (a, f(a)).(B) The tangent line intersects the x-axis.(C) The tangent line intersects the f(x)-axis (the y-axis).(D) Any line through (a, f(a)) other than the tangent line intersects the graph of f at at least oneother point.(E) None of the above need b e true.Your answer:(4) Which of the following verbal statements is not valid as a general rule?(A) The derivative of the sum of two functions is the sum of the derivatives of the functions.(B) The derivative of the difference of two functions is the difference of the derivatives of the func-tions.(C) The derivative of a constant times a function is the same constant times the derivative of thefunction.(D) The derivative of the product of two functions is the product of the derivatives of the functions.(E) None of the above, i.e., they are all valid as general rules.Your