Homework

Problem 1 · easy

Evaluate lim x->4 (2x + 3) by direct substitution.

Problem 2 · easy

Evaluate lim x->-1 (x^2 - 3x + 1) by direct substitution.

Problem 3 · medium

Use a table to estimate lim x->0 (sin(x)/x). (Use x = 0.1, 0.01, 0.001 in radians.)

Problem 4 · easy

Evaluate lim x->3 (x^2 - 9)/(x - 3).

Problem 5 · easy

Evaluate lim x->-1 (x^2 + 2x + 1)/(x + 1).

Problem 6 · medium

Evaluate lim x->2 (x^3 - 8)/(x - 2).

Problem 7 · easy

Evaluate lim x->infinity (5x^2 - 3)/(2x^2 + 1).

Problem 8 · easy

Evaluate lim x->infinity (4x + 1)/(x^2 - 3).

Problem 9 · medium

Evaluate lim x->infinity (x^3 - 2x)/(3x^2 + 1).

Problem 10 · medium

Is f(x) = (x^2-1)/(x-1) continuous at x=1? Explain.

Problem 11 · medium

For f(x) = {x^2 if x<1; 2x-1 if x>=1}, is f continuous at x=1? Show the check.

Problem 12 · medium

For f(x) = {x+3 if x<2; x^2 if x>2} and f(2) undefined, classify the discontinuity at x=2.

Problem 13 · medium

Find the average rate of change of f(x) = x^2 + 1 from x=1 to x=1+h, and state what it approaches as h->0.

Problem 14 · challenge

Find the instantaneous rate of change (limit) of f(x) = 3x^2 at x=2.

Problem 15 · challenge

Evaluate lim x->0 (sqrt(x+4) - 2)/x by multiplying by the conjugate.

Problem 16 · easy

Factor and simplify: (x^2 + x - 6)/(x - 2).

Problem 17 · medium

What is lim x->0 (1-cos(x))/x? (State the answer — this is a standard calculus limit.)

Problem 18 · medium

Evaluate lim x->3 (x^2 - 5x + 6)/(x^2 - x - 6).

Problem 19 · easy

Name three skills from Pre-Calculus that you will use every week in Calculus I.

Problem 20 · easy

Evaluate lim x->pi sin(x).

Problem 21 · easy

State the three conditions for a function to be continuous at x=a.

Problem 22 · challenge

Evaluate lim x->1 (x^n - 1)/(x - 1) for positive integer n.

Problem 23 · medium

Find the average rate of change of f(x) = sin(x) from x=0 to x=π/2.

Problem 24 · medium

What does lim h->0 (f(x+h)-f(x))/h represent in calculus?

Problem 25 · challenge

Set up the limit definition of the derivative for f(x) = x^2 at x=3 and evaluate it.