Name: Bekenov Kuandyk
1. Prove that fx=2x+5 is one-to-one function. (10 points)
Let make an assumption tha fx1=fx2t and then prove that x1=x2.
Then fx=2x+5 is one-to-one function
Let f: A→B, as given below. Is f a one-to-one function? Please explain why or why not. (10 points)
F is not one-to-one function, because f(1)=1 and f(2)=1.
3. The modulo function (a mod n or a modulo n) maps every positive integer number to the remainder of the division of a/n. For example, the expression 22 mod 5 would evaluate to 2 since 22 divided by 5 is 4 with a remainder of 2. The expression 10 mod 5 would resolve to 0 since 10 is divisible by 5 and there is not a remainder.
a. If n is fixed ...view middle of the document...
Find lim2x99+x+1,000,000x100+2,000,000 as x→0 (6 points)
8. Prove that fx=x2+7 is a continuous function. (8 points)
Let prove that fx+∆x-fx→0 when ∆x→0 then:
x+∆x2+7-x2+7=x2+2x∆x+∆x2+7-x2-7-=∆x2x+∆x=0, it means fx= x2+7 is a continuous function.
9. Find derivatives of the following functions using differentiation rules. (Do not use the definition of a derivative!)
c. fx=2x+5 (3 points)
d. fx=2x5+x3-6x+100 (3 points)
e. fx=ex+2(4x-1) (3 points)
f. fx=7ex (3 points)
g. fx=x2(x-1) (3 points)
h. fx=3x2+7 (3 points)
i. fx=ex(2x+5) (3 points)
j. fx=ex+2+4x-1 (3 points)
10. Find the derivative of the following function at :fx=2x3+x2+9 (8 points)
11. Find the derivative of the following function fx=2x2-x using the definition of a derivative. (Hint: though you can use the rules of differentiation to check your answer, you must use the definition of a derivative to solve this problem in order to receive any credit for your response) (10 points)
Complementary problems (worth up to 3 extra credit points each)
1. Let fx=2x and gx=ex, where. Find and as well as the domain and range of these functions.
domain of these functions: x∈-∞,∞
rangeof these functions: any real power of any positive number is always positive, whether the exponent is positive, negative or 0.So the range is “all positive numbers”,
or in interval notation (0, ∞)
2. Find the derivative of by using the definition of the derivative