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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 27a
Chapter 3, Problem 27a

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = x3; P (1,1)

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Step 1: Recall the definition of the derivative as the slope of the tangent line at a point. The derivative of a function f at a point x = a is given by the limit: \( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \).
Step 2: Identify the function and the point of interest. Here, the function is \( f(x) = x^3 \) and the point P is (1, 1). We need to find \( f'(1) \).
Step 3: Substitute the function into the derivative definition. Calculate \( f(1+h) = (1+h)^3 \) and \( f(1) = 1^3 = 1 \).
Step 4: Expand \( (1+h)^3 \) using the binomial theorem: \( (1+h)^3 = 1 + 3h + 3h^2 + h^3 \).
Step 5: Substitute these into the derivative formula: \( f'(1) = \lim_{h \to 0} \frac{(1 + 3h + 3h^2 + h^3) - 1}{h} \). Simplify the expression and evaluate the limit as \( h \to 0 \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point, which is crucial for understanding how the function behaves locally.
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Derivative

The derivative of a function at a point quantifies how the function's output changes as its input changes. It is defined as the limit of the average rate of change of the function as the interval approaches zero. For the function f(x) = x^3, the derivative can be calculated using the power rule.
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Power Rule

The power rule is a basic differentiation rule that states if f(x) = x^n, then f'(x) = n*x^(n-1). This rule simplifies the process of finding derivatives for polynomial functions, making it essential for calculating the slope of the tangent line for functions like f(x) = x^3.
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Related Practice
Textbook Question

Suppose a baseball is thrown vertically upward from the ground with an initial velocity of v0 ft/s. The approximate height of the ball (in feet) above the ground after t seconds is given by s(t) = -16t² + v0t.

With what velocity does the ball strike the ground?

Textbook Question

Evaluate the derivative of the following functions.

f(t) = ln (sin-1 t2)

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Textbook Question

Find and simplify the derivative of the following functions.

y = (3t−1)(2t−2)-1

Textbook Question

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x)= 1/(2x + 1); P (0,1)

Textbook Question

Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 64 ft/s from a height of 32 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t2 + 64t + 32.

On what intervals is the speed increasing?

Textbook Question

Evaluate the derivative of the following functions.

f(x) = x2 + 2x3 cot-1 x - ln (1 + x2)