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Calculus: Early Transcendental Functions

Robert Smith, Roland Minton

Chapter 1

Limits and Continuity - all with Video Answers

Educators

Section 1

A Brief Preview of Calculus: Tangent Lines and the Length of a Curve

01:02

Problem 1

Estimate the slope (as in example 1.1) of $y=f(x)$ at $x=a$
$f(x)=x^{2}+1$
(a) $a=1$
(b) $a=2$

Carson Merrill
Carson Merrill
Numerade Educator
01:06

Problem 2

Estimate the slope (as in example 1.1) of $y=f(x)$ at $x=a$
$f(x)=x^{3}+2$
(a) $a=1$
(b) $a=2$

Carson Merrill
Carson Merrill
Numerade Educator
01:02

Problem 3

Estimate the slope (as in example 1.1) of $y=f(x)$ at $x=a$
$f(x)=\cos x$
(a) $a=0$
(b) $a=\pi / 2$

Carson Merrill
Carson Merrill
Numerade Educator
01:05

Problem 4

Estimate the slope (as in example 1.1) of $y=f(x)$ at $x=a$
$f(x)=\sqrt{x+1}$
(a) $a=0$
(b) $a=3$

Carson Merrill
Carson Merrill
Numerade Educator
01:03

Problem 5

Estimate the slope (as in example 1.1) of $y=f(x)$ at $x=a$
$f(x)=e^{x}$
(a) $a=0$
(b) $a=1$

Carson Merrill
Carson Merrill
Numerade Educator
01:02

Problem 6

Estimate the slope (as in example 1.1) of $y=f(x)$ at $x=a$
$f(x)=\ln x$
(a) $a=1$
(b) $a=2$

Carson Merrill
Carson Merrill
Numerade Educator
01:04

Problem 7

Estimate the length of the curve $y=f(x)$ on the given interval using (a) $n=4$ and (b) $n=8$ line segments. c) If you can program a calculator or computer, use larger $n$ 's and conjecture the actual length of the curve.
$$
f(x)=\cos x, 0 \leq x \leq \pi / 2
$$

Carson Merrill
Carson Merrill
Numerade Educator
01:03

Problem 8

Estimate the length of the curve $y=f(x)$ on the given interval using (a) $n=4$ and (b) $n=8$ line segments. c) If you can program a calculator or computer, use larger $n$ 's and conjecture the actual length of the curve.
$$
f(x)=\sin x, 0 \leq x \leq \pi / 2
$$

Carson Merrill
Carson Merrill
Numerade Educator
01:07

Problem 9

Estimate the length of the curve $y=f(x)$ on the given interval using (a) $n=4$ and (b) $n=8$ line segments. c) If you can program a calculator or computer, use larger $n$ 's and conjecture the actual length of the curve.
$$
f(x)=\sqrt{x+1}, 0 \leq x \leq 3
$$

Carson Merrill
Carson Merrill
Numerade Educator
01:03

Problem 10

Estimate the length of the curve $y=f(x)$ on the given interval using (a) $n=4$ and (b) $n=8$ line segments. c) If you can program a calculator or computer, use larger $n$ 's and conjecture the actual length of the curve.
$$
f(x)=1 / x, 1 \leq x \leq 2
$$

Carson Merrill
Carson Merrill
Numerade Educator
02:32

Problem 11

Estimate the length of the curve $y=f(x)$ on the given interval using (a) $n=4$ and (b) $n=8$ line segments. c) If you can program a calculator or computer, use larger $n$ 's and conjecture the actual length of the curve.
$$
f(x)=x^{2}+1,-2 \leq x \leq 2
$$

Audrey Fong
Audrey Fong
Numerade Educator
02:12

Problem 12

Estimate the length of the curve $y=f(x)$ on the given interval using (a) $n=4$ and (b) $n=8$ line segments. c) If you can program a calculator or computer, use larger $n$ 's and conjecture the actual length of the curve.
$$
f(x)=x^{3}+2,-1 \leq x \leq 1
$$

Audrey Fong
Audrey Fong
Numerade Educator
01:09

Problem 13

Sketch the parabola $y=1-x^{2}$ and shade in the region above the $x$ -axis between $x=-1$ and $x=1$. (a) Sketch in the following rectangles: (1) height $f\left(-\frac{3}{4}\right)$ and width $\frac{1}{2}$ extending from $x=-1$ to $x=-\frac{1}{2} .$ (2) height $f\left(-\frac{1}{4}\right)$ and width $\frac{1}{2}$ extending from $x=-\frac{1}{2}$ to $x=0 .$ (3) height $f\left(\frac{1}{4}\right)$ and width $\frac{1}{2}$ extending from $x=0$ to $x=\frac{1}{2}$. (4) height $f\left(\frac{3}{4}\right)$ and width $\frac{1}{2}$ extending from $x=\frac{1}{2}$ to $x=1 .$ Compute the sum of the areas of the rectangles. (b) Divide the interval [-1,1] into 8 pieces and construct a rectangle of the appropriate height on each subinterval. Find the sum of the areas of the rectangles. Compared to the approximation in part (a), explain why you would expect this to be a better approximation of the actual area under the parabola.

Carson Merrill
Carson Merrill
Numerade Educator
18:55

Problem 14

Use a computer or calculator to compute an approximation of the area in exercise 13 using (a) 16 rectangles, (b) 32 rectangles, (c) 64 rectangles. Use these calculations to conjecture the exact value of the area under the parabola.

Yuki Hotta
Yuki Hotta
Numerade Educator
01:09

Problem 15

Use the technique of exercise 13 to estimate the area below $y=\sin x$ and above the $x$ -axis between $x=0$ and $x=\pi$

Carson Merrill
Carson Merrill
Numerade Educator
01:02

Problem 16

Use the technique of exercise 13 to estimate the area below $y=x^{3}$ and above the $x$ -axis between $x=0$ and $x=1$

Carson Merrill
Carson Merrill
Numerade Educator
01:05

Problem 17

Estimate the length of the curve $y=\sqrt{1-x^{2}}$ for $0 \leq x \leq 1$ with (a) $n=4$ and (b) $n=8$ line segments. Explain why the exact length is $\pi / 2 .$ How accurate are your estimates?

Carson Merrill
Carson Merrill
Numerade Educator
01:07

Problem 18

Estimate the length of the curve $y=\sqrt{9-x^{2}}$ for $0 \leq x \leq 3$ with (a) $n=4$ and (b) $n=8$ line segments. Explain why the exact length is $3 \pi / 2$. How would an estimate of $\pi$ from part (b) of the exercise compare to that obtained in part (b) of exercise $17 ?$

Carson Merrill
Carson Merrill
Numerade Educator