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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 94a
Chapter 3, Problem 94a

Tangent lines Assume f is a differentiable function whose graph passes through the point (1, 4). Suppose g(x)=f(x²) and the line tangent to the graph of f at (1, 4) is y=3x+1. Find each of the following.
a. g(1)

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1
Step 1: Understand the problem. We are given a function g(x) = f(x^2) and need to find g(1). We know that f is differentiable and passes through the point (1, 4), and the tangent line to f at (1, 4) is y = 3x + 1.
Step 2: Evaluate g(1). Since g(x) = f(x^2), we substitute x = 1 into g(x) to get g(1) = f(1^2) = f(1).
Step 3: Use the information about f. We know that the graph of f passes through the point (1, 4), which means f(1) = 4.
Step 4: Conclude the evaluation. Since f(1) = 4, we have g(1) = f(1) = 4.
Step 5: Summarize the result. Therefore, g(1) is equal to 4.

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Key Concepts

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

Differentiable Functions

A differentiable function is one that has a derivative at every point in its domain. This means that the function is smooth and continuous, allowing for the calculation of slopes of tangent lines at any point. The existence of a derivative indicates that the function can be locally approximated by a linear function, which is essential for understanding tangent lines.
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Finding Differentials

Tangent Lines

A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve at that point. The equation of the tangent line can be derived using the point-slope form, which incorporates the derivative of the function at that point. In this case, the tangent line to f at (1, 4) is given as y = 3x + 1, indicating that the slope at that point is 3.
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Slopes of Tangent Lines

Composition of Functions

The composition of functions involves combining two functions where the output of one function becomes the input of another. In this problem, g(x) = f(x²) represents a composition where the input x is squared before being passed to function f. Understanding how to evaluate g(1) requires substituting 1 into the composition and then determining the value of f at the resulting input.
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Evaluate Composite Functions - Special Cases
Related Practice
Textbook Question

Composition containing sin x Suppose f is differentiable on [−2,2] with f′(0)=3 and f′(1)=5. Let g(x)=f(sin x). Evaluate the following expressions.

c. g'(π)

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Textbook Question

Suppose the line tangent to the graph of f at x=2 is y=4x+1 and suppose y=3x−2 is the line tangent to the graph of g at x=2. Find an equation of the line tangent to the following curves at x=2.

y = f(x)g(x)

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Textbook Question

{Use of Tech} Beak length The length of the culmen (the upper ridge of a bird’s bill) of a t-week-old Indian spotted owlet is modeled by the function L(t)=11.94 / 1 + 4e^−1.65t, where L is measured in millimeters.


a. Find L′(1) and interpret the meaning of this value.

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Textbook Question

Derivatives from tangent lines Suppose the line tangent to the graph of f at x=2 is y=4x+1 and suppose y=3x−2 is the line tangent to the graph of g at x=2. Find an equation of the line tangent to the following curves at x=2.

b. y = f(x) / g(x)

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Textbook Question

Given that p(x) = (5e^x+10x⁵+20x³+100x²+5x+20) ⋅ (10x⁵+40x³+20x²+4x+10), find p′(0) without computing p′(x).

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Textbook Question

Composition containing sin x Suppose f is differentiable for all real numbers with f(0)=−3,f(1)=3,f′(0)=3, and f′(1)=5. Let g(x)=sin(πf(x)). Evaluate the following expressions.

b. g'(1)

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