Textbook Question
Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.
f(x) = √(3x + 3); P(2,3)
223
views
2. Intro to Derivatives
Tangent Lines and Derivatives
2:01 minutesProblem 3.33a
Textbook Question
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = 8x; a = −3
Verified step by step guidance1
Step 1: Identify the function f(x) = 8x and the point a = -3 where you need to find the derivative.
Step 2: Recall that the derivative of a linear function f(x) = mx is f'(x) = m. In this case, m = 8.
Step 3: Since the derivative of f(x) = 8x is constant, f'(x) = 8 for all x.
Step 4: Evaluate the derivative at the given point a = -3. Since f'(x) = 8 for all x, f'(-3) = 8.
Step 5: Conclude that the derivative of the function at the point a = -3 is f'(-3) = 8.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate of change of a function with respect to its variable. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In practical terms, the derivative at a point gives the slope of the tangent line to the function at that point.
Recommended video:
05:44
Derivatives
Tangent Lines
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 is equal to the derivative of the function at that point. This concept is crucial for understanding how functions behave locally around specific values.
Recommended video:
05:13Slopes of Tangent Lines
Function Evaluation
Function evaluation involves substituting a specific value into a function to determine its output. In the context of derivatives, evaluating the function at a point helps in calculating the derivative at that point. For example, in the function f(x) = 8x, evaluating at a = -3 allows us to find the slope of the tangent line at that specific x-value.
Recommended video:
4:26Evaluating Composed Functions
Related Videos
Related Practice