# Math 1251 Practice T-F solutions

1. The equation of the tangent line to the curve y=f(x) at the point (a,f(a)) is y-f(a)=f'(x)(x-a).

FALSE. Take f(x)=x and a=0. The tangent line is y=x, but the given equn is y-0=x(x-0), not a line. We need f'(a) not f'(x), which is the slope of the tangent line at x=a.

2. The derivative with respect to x of f(x)/g(x) is f'(x)/g'(x).

FALSE. Take f(x)=g(x)=x, then the derivative of x/x=1 is 0, while the quotient f'(x)/g'(x)=1/1=1. We use the quotient rule for a correct answer.

3. If f(0)=0, then f'(0)=0.

FALSE. Take f(x)=x which has f(0)=0 but f'(0)=1.

4. If f(x) is differentiable at x=2, then f(x) is continuous at x=2.

TRUE. Differentiability at a point x=a always implies continuity at that same point.

5. If f(x) approaches infinity as x approaches 0, then x*f(x) also approaches infinity as x approaches 0.

FALSE. Let f(x)=1/x. Then f(x)-> infty, but x*f(x)=1->1 as x->0.

6. If lim_{x->2} f(x)=4, and lim_{x->3} g(x)=7, then lim_{x->5} (f(x)+g(x))=11.

FALSE. The values of f(x) near 2 and g(x) near 3 say nothing at all about the values of f(x)+g(x) near 5.

7. The derivative with respect to x of \pi^6 is 6*\pi^5.

FALSE. \pi^6 is a constant, so the derivative =0.

8. If f(x)->infinity, g(x)->infinity as x->infinity, then f(x)/g(x) could either ->infinty, ->0, or go to any constant as x->infinity.

TRUE. f(x)=x^2, and g(x)=x has f(x)/g(x)->infty. f(x)=2x, g(x)=x, has f(x)/g(x)->2 as x-> infty, and clearly we could replace 2 by any constant. The same reasoning shows f(x)/g(x)->0 is possible.

9. There is no function which grows more rapidly than f(x)=e^x.

FALSE. For example g(x)=e^(x^2} has the property that as x-> infty, f(x)/g(x)->0.

10. If f(x) has a local mimimum at x=2, then f'(2)=0.

FALSE. f(x) may not even have a derivative at x=2, for exmaple, f(x)=|x-2|. If f is diff at x=2 then the derivative is =0.

11. If f''(x) exists and is >0 for all x, then f(x) is an increasing function of x.

FALSE. If f(x)=x^2, then f''(x)=2>0 for all x, yet x^2 is decreasing for x<0.

12. The derivative with respect to x of x^x is x*x^{x-1}.

FALSE.We cannot use the power rule, since the exponent depends upon x. The correct answer is to write x^x=e^(xlog(x)) and then use the chain rule.

13. If f(1)=2, and f(3)=-3, and f(x) is continuous on [1,3], then f(a)=0 for some a in (1,3).

TRUE. By the intermediate value theorem, since f in continuous on a closed bounded interval, f must attain all values between -3 and 2, so it f(a)=0 for some a in (1,3).

14. If f'(x)->0 as x->infinity, then we must also have f(x)-> a constant c as x->infinity.

FALSE. Let f(x)=log(x), then f'(x)=1/x->0 as x->infty, yet f(x) does not approach a constant, f(x) -> infty as x-> infty.

15. If f(x) has a local minimum on the interval (1,2), then the absolute minimum of f(x) on [1,2] cannot occur at x=1.

FALSE. f(x) could increase starting from x=1, have local max, then decrease to x=1/2, have a local min there, then increase to x=2. If the local min at x=1/2 is greater than the endpoint value at x=1, then the absolute minimum does occur at x=1. (Draw a picture.)

16. If f(5)=2, f(7)=9, then f'(6)=7/2.

FALSE. The Mean Value Theorem says there is some c between 5 and 7 with f'(c)=(9-2)/(7-5), c does not have to be in middle of the interval.FOr exmaple, f(x)=7((x-5)/2)^3+2 has f(5)=2, f(7)=9, but f'(6)=21/8.

17. The best linear approximation to e^(2x) at x=0 is 1+2x.

TRUE. If f(x)=e^(2x), f(0)=1, f'(0)=2, so the tangent line at x=0 is y=1+2x, which is the best linear approx at x=0.

18. If f(x) is differentiable at x=2, and f(x) has a local maximum at x=2, then f'(2)=0.

TRUE. This is one of our theorems, that differentiability plus local max or min implies the derivative there is 0.

19. If f(x) is continuous on [2,4], then f(x) must have an absolute maximum on [2,4].

TRUE. Since the interval [2,4] is closed and bounded, f must have an absolute maximum there.

20. If f(x) and g(x) both have local maxima at x=2, then f(x)*g(x) also has a local maximum at x=2.

FALSE. (IF f(x) and g(x) are postive this would be true.) Let f(x)=g(x)=-x^2. Then f(x) and g(x) have local maxima at x=0, but f(x)*g(x)=x^4 has a local minimum at x=0.

21. The 12th derivative with respect to x of any polynomial in x of degree 10 is 0.

TRUE. The derivative reduces the degree of a polynomial by 1, so degree 10 becomes degree 0 after 11 derivatives, that is 0.

22. If f(x)=f'(x) for all x, then we also have that f''(x)=f(x) for all x.

TRUE. Just diferentiate both sides to get f'(x)=f''(x), and we know f'(x)=f(x).

23. The function y=log(x) is increasing and concave downward for x<0.

TRUE. dy/dx=1/x, d^2y/dx^2=-1/x^2<0 so log(x) is increasing and concave downward.

24. If f'(0) does not exist, then lim_{x->0} f(x) does not exist.

FALSE. f(x)=|x| is not diff at x=0, yet lim_{x->0} f(x)=0 exists.

25. The nth derivative of e^(x) is e^x.

TRUE. Since the first derviative of e^x is e^x, any successive derivatives also give e^x.

26. The function f(x)=2-x^2 on the real line is a 1-1 function.

FALSE. f(1)=f(-1)=1, so two different numbers are mapped to 1 by f.

27. For any cubic polynomial p(x) with real coefficients there is a real number x such that p(x)=0.

TRUE. Imagine the graph of y=p(x) for |x| large. Suppose that the cubic term is c*x^3, c not equal to 0. If c>0, then for large x, the largest term in p(x) is c*x^3, so p(x)>0 for large x. Similarly, for x->-\infty, p(x)<0. By the intermediate value theorem, since p(x) is continuous, p(x) must attain all real values, including 0. The identical proof works for c<0.

28. If f(x) approaches 0 when x-> \infinity, then we must also have that f'(x) approaches 0 as x-> \infinity.

FALSE. f(x) could have many spiky tiny wiggles, that make f'(x) big, yet f(x) is tiny. For example, f(x)=sin(e^x)/x.

29. If f''(0)=0, then x=0 must be a point of inflection of the graph y=f(x).

FALSE. f(x)=x^4, has f''(0)=0, yet f(x) is concave upward for all x.

30. If f(x) is a 1-1 function with domain the real numbers, then f'(0) cannot be equal to 0.

FALSE. f(x)=x^3 is 1-1, and f'(0)=0.

31. If p(x) is a polynomial in x, then p'(x) is a polynomial in x.

TRUE. Taking the derivative reduces the degree of a polynomial by one, but keeps it a polynomial.

32. If f(x) is continuous on the interval [1,3], f(x)>=0, f(2)=0, then f has a local minima at x=2.

TRUE. The values of f near 2 are >=0=f(2).

33. The nth derivative of ln(x) is 1/x^n.

FALSE. There is a minus sign and factorial missing, (-1)^{n-1}*(n-1)!/x^n.

34. If p(x) is a polynomial with p(2)=5=p(3), then there is some number c, 3>c>2, with p'(c)=0.

TRUE. This follows from Rolle's theorem.

35. There is some function f(x) such that f(x)>0, f'(x)<0, f''(x)>0 for all x.

TRUE. Imagine a graph which is above the x-axis, decreasing, yet concave upward, such as y=e^(-x).