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Limits and derivatives exercises · Physics Playground
Exercises
Math
Limits and derivatives
29 problems with worked solutions, from Precalculus.
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Rates of Change: Derivatives
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Precalculus
12.1 Finding Limits: Numerical and Graphical Approaches
Example 1
Understanding the Limit of a Function
For the following limit, define a,f(x), and L. lim x→2 ( 3x+5)=11
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Example 2
Finding a Limit Using a Graph
Determine the following limits and function value for the function f shown in Figure 6. lim x→2^− f(x) lim x→2^+ f(x) lim x→2 f(x) f(2) Figure 6 Determine the following limits and function value for the function f…
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Example 3
Finding a Limit Using a Table
Numerically estimate the limit of the following expression by setting up a table of values on both sides of the limit. lim x→0 ( (5sin(x))/(3x))
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Example 4
Using a Graphing Utility to Determine a Limit
With the use of a graphing utility, if possible, determine the left- and right-hand limits of the following function as x approaches 0. If the function has a limit as x approaches 0, state it. If not, discuss why…
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Precalculus
12.2 Finding Limits: Properties of Limits
Example 1
Evaluating the Limit of a Function Algebraically
Evaluate lim x→3 ( 2x+5).
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Example 2
Evaluating the Limit of a Function Algebraically
Evaluate lim x→3 ( 5x^2).
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Example 3
Evaluating the Limit of a Polynomial Algebraically
Evaluate lim x→5 ( 2x^3 −3x+1).
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Example 4
Evaluating a Limit of a Power
Evaluate lim x→2 ( 3x+1) ^5.
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Example 5
Evaluating the Limit of a Quotient by Factoring
Evaluate lim x→2 ( (x^2 −6x+8)/(x−2)).
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Example 6
Evaluating the Limit of a Quotient by Finding the LCD
Evaluate lim x→5 ( (1/x −1/5)/(x−5)).
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Example 7
Evaluating a Limit Containing a Root Using a Conjugate
Evaluate lim x→0 ( (√(25−x) −5)/x).
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Example 8
Evaluating the Limit of a Quotient of a Function by Factoring
Evaluate lim x→4 ( (4−x)/(√x −2)).
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Example 9
Evaluating the Limit of a Quotient with Absolute Values
Evaluate lim x→7 (| x−7 |)/(x−7).
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Precalculus
12.3 Continuity
Example 1
Identifying Discontinuities
Identify all discontinuities for the following functions as either a jump or a removable discontinuity. ⓐ f(x)=(x^2 −2x−15)/(x−5) ⓑ g(x)={ x+1, x<2 −x, x≥2
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Example 2
Determining Whether a Piecewise Function is Continuous at a Given Number
Determine whether the function f(x)={ 4x, x≤3 8+x, x>3 is continuous at ⓐ x=3 ⓑ x=8/3
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Example 3
Determining Whether a Rational Function is Continuous at a Given Number
Determine whether the function f(x)=(x^2 −25)/(x−5) is continuous at x=5.
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Example 4
Determining the Input Values for Which a Piecewise Function Is Discontinuous
Determine whether the function f is discontinuous for any real numbers. f(x)={ x+1, x<2 3, 2≤x<4 x^2 −11, x≥4
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Example 5
Determining Whether a Piecewise Function Is Continuous
Determine whether the function below is continuous. If it is not, state the location and type of each discontinuity. f(x)={ sin(x), x<0 x^3, x>0
Precalculus
12.4 Derivatives
Example 1
Finding the Average Rate of Change
Find the average rate of change connecting the points ( 2,−6) and ( −1,5).
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Example 2
Finding the Derivative of a Polynomial Function
Find the derivative of the function f(x)=x^2 −3x+5 at x=a.
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Example 3
Finding the Derivative of a Rational Function
Find the derivative of the function f(x)=(3+x)/(2−x) at x=a.
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Example 4
Finding the Derivative of a Function with a Root
Find the derivative of the function f(x)=4√x at x=36.
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Example 5
Finding the Instantaneous Rate of Change
Using the function above, s(t)=−16t^2 +64t+6, what is the instantaneous velocity of the ball at 1 second and 3 seconds into its flight?
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Example 6
Estimating the Derivative at a Point on the Graph of a Function
From the graph of the function y=f( x) presented in Figure 5, estimate each of the following: ⓐ f(0) ⓑ f(2) ⓒ f'(0) ⓓ f'(2) Figure 5
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Example 7
Finding a Marginal Cost
The cost in dollars of producing x laptop computers in dollars is f( x)=x^2 −100x. At the point where 200 computers have been produced, what is the approximate cost of producing the 201st unit?
Example 8
Interpreting a Derivative in Context
A car leaves an intersection. The distance it travels in miles is given by the function f( t), where t represents hours. Explain the following notations: ⓐ f(0)=0 ⓑ f^′ (1)=60 ⓒ f(1)=70 ⓓ f(2.5)=150
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Example 9
Determining Where a Function Is Continuous and Differentiable from a Graph
Using Figure 15, determine where the function is continuous discontinuous differentiable not differentiable At the points where the graph is discontinuous or not differentiable, state why. Figure 15
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Example 10
Finding the Equation of a Line Tangent to a Function at a Point
Find the equation of a line tangent to the curve f(x)=x^2 −4x at x=3.
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Example 11
Finding the Instantaneous Velocity
A ball is tossed upward from a height of 200 feet with an initial velocity of 36 ft/sec. If the height of the ball in feet after t seconds is given by s(t)=−16t^2 +36t+200, find the instantaneous velocity of the ball…
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