Calculus

1. Evaluate the function at the given numbers (correct to six decimals places). Use the results to guess the value of the limit,or explain why it does not exist.

F(t)=( t^(1/3) - 1)/(t^(1/2) - 1) ;
t= 1.5,1.2,1.1,1.01,1.001;
The limit of F(t) as t approaches 1.

My work:

t=1.001, F(t)= 0.666611
t=1.01, F(t)=0.666114
t=1, F(t)= empty
t= 1.1, F(t)=0.661358
t=1.2,F(t)= 0.656488
t=1.5,F(t)=0.643905

The limit of F(t) as x approaches 1 is 0.66????

2. The slope of the tangent line to the graph of the exponential function y=2^x at the point (0,1) is the limit of (2^x-1)/x as x approaches 0. Estimate the slope to three decimal places.

My answer: 0.597?????

  1. 👍 0
  2. 👎 0
  3. 👁 1,011
  1. #1 using l'Hospital's Rule,

    ( t^(1/3) - 1)/(t^(1/2) - 1) -> (1/3 t^2/3)/(1/2 t^1/2)
    = 2/3 t^-1/6
    = 2/3 at t=1

    If y=2^x,
    y' = ln2 2^x
    at x=0, that's just ln2
    So, the line is

    y-1 = ln2(x)
    so, y = ln2 x + 1
    So, the slope is ln2 = 0.693
    See

    http://www.wolframalpha.com/input/?i=plot+y%3D2^x%2C+y%3Dlog2+*+x+%2B+1

    Or, you can evaluate the limit, but you wind up with the same value.

    What steps got you to 0.597?

    1. 👍 0
    2. 👎 0

Respond to this Question

First Name

Your Response

Similar Questions

  1. Calc

    Find the linear approximation of the function g(x) = fifth root(1 + x) at a = 0. g(x)≈ 1+(x/5) Correct: Your answer is correct. Use it to approximate the numbers fifth root (0.95) and fifth root (1.1) (Round your answers to

    asked by Anonymous on May 11, 2018
  2. Algebra plz help

    1.Identify the domain and range of the following relation. {(3,7),(3,8),(3,-2),(3,4),(3,1)} A.Domain:{3} Range:{-2,1,4,7,8}**** B.Domain:{-2,1,4,7,8} Range:{3} C.Domain:{all real numbers} Range:{all real numbers} D.Domain:O/

    asked by GummyBears on October 20, 2015
  3. Mathematics

    When two balanced dice are rolled, 36 equally likely outcomes are possible as shown below. (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (4, 1) (4, 2)

    asked by Jamie on December 5, 2009
  4. ALGEBRA

    A single card is selected from an ordinary deck of cards. The sample space is shown in the following figure. Find the following probabilities. (Enter the answers either as fractions or as decimals rounded to three places.) P(five

    asked by Mike on May 23, 2012
  5. Algebra

    1.Identify the domain and range of the following relation. {(3,7),(3,8),(3,-2),(3,4),(3,1)} A.Domain:{3} Range:{-2,1,4,7,8}**** B.Domain:{-2,1,4,7,8} Range:{3} C.Domain:{all real numbers} Range:{all real numbers} D.Domain:O/

    asked by GummyBears on October 19, 2015
  1. MATH

    Consider F and C below. F(x, y, z) = 2xz + y2 i + 2xy j + x2 + 15z2 k C: x = t2, y = t + 2, z = 3t − 1, 0 ≤ t ≤ 1 (a) Find a function f such that F = ∇f. f(x, y, z) = x2z+xy2+5z3+C Correct: Your answer is correct. (b) Use

    asked by Anon on November 9, 2016
  2. Math

    Use a calculator to evaluate the trigonometric function. Round to four decimal places. Make sure that your calculator is in the correct angle mode. 1) csc 2pi/5 1.3213? 2) sec 0.7 1.3075?

    asked by Anonymous on September 9, 2015
  3. Maths

    Use the binomial theorem to evaluate (1.004)^8 correct to 5 decimal places.

    asked by Ande2 on June 27, 2019
  4. math

    Find the absolute extrema of the function on the closed interval. Use a graphing utility to verify your results. (Round your answers to two decimal places. If an answer does not exist, enter DNE.) g(x) = (x^2 − 4)^2/3, [−5, 3]

    asked by akash on March 31, 2013
  5. math check answers

    For questions 3 and 4, you will be answering by filling in the blank. Please be aware that your answer must include any commas or decimals in their proper places in order to be correct. The dollar signs have been provided. For

    asked by megan on February 10, 2017
  6. Math

    The function b(h) models the percent of a certain wildflower that blooms during months in which the average daily sunlight is h hours. Name the most appropriate domain for this function. A. Integers, h ≤ 24 B. Integers, 0 ≤ h

    asked by Sanjay Gupta on May 1, 2016

You can view more similar questions or ask a new question.