Question

In the study of Differential Calculus, in general, the first application of limit that we learn allows us to obtain the equation of the tangent line to the graph of a given function f at a point P(x,y). This concept known as Derivative has numerous applications that allow us to measure rates of variation, such as speed, acceleration, growth, decrease, etc. Thus, consider f(x) = x2 a real function defined on every line R, determine the equation of the tangent line to this curve at the point P(2,4).

277

likes
1387 views

Answer to a math question In the study of Differential Calculus, in general, the first application of limit that we learn allows us to obtain the equation of the tangent line to the graph of a given function f at a point P(x,y). This concept known as Derivative has numerous applications that allow us to measure rates of variation, such as speed, acceleration, growth, decrease, etc. Thus, consider f(x) = x2 a real function defined on every line R, determine the equation of the tangent line to this curve at the point P(2,4).

Expert avatar
Fred
4.4
111 Answers
1. Given the function f(x) = x^2 and a point P(2, 4), we need to find the derivative of f(x) using the limit definition:

f'(x) = \lim_{{h \to 0}} \frac{{f(x + h) - f(x)}}{h}

2. Substitute f(x) = x^2:

f'(x) = \lim_{{h \to 0}} \frac{{(x + h)^2 - x^2}}{h}

3. Simplify the expression inside the limit:

f'(x) = \lim_{{h \to 0}} \frac{{x^2 + 2xh + h^2 - x^2}}{h}

f'(x) = \lim_{{h \to 0}} \frac{{2xh + h^2}}{h}

4. Factor out h:

f'(x) = \lim_{{h \to 0}} \frac{{h(2x + h)}}{h}

f'(x) = \lim_{{h \to 0}} (2x + h)

5. As h approaches 0:

f'(x) = 2x

6. Find the slope at x = 2:

f'(2) = 2(2) = 4

7. Use the point-slope form of the equation of a line with slope m = 4 and point (2, 4):

y - 4 = 4(x - 2)

8. Simplify to the final equation:

y - 4 = 4x - 8

y = 4x - 4

Answer: y = 4x - 4

Frequently asked questions (FAQs)
What are the characteristics of the ellipse function when the major axis is vertical?
+
How many gallons of water will be needed if each person drinks 8 ounces of water daily for 30 days?
+
Question: What are the basic shapes of the graphs of quadratic, linear, and exponential functions?
+
New questions in Mathematics
10! - 8! =
CASE 6-1: PREPARE A PRODUCTION PLAN: WHAT PROBLEMS ARRIVE? Midwest Plastics Company has conducted profit planning for several years. The president stated (with justification) that inventory control and planning had not been satisfactory, which was mainly due to poor planning of production and inventory budgets. Please analyze and provide recommendations, in detail, on the issue regarding the 20B profit plan, which is now being prepared. Their analysis and recommendations will be presented to the executive committee. Despite the seasonality factor, the sales department has been successful in developing a sales plan, on a monthly basis, for each year. The following sales data is available for 20B. 1. Sales plan summary for 20B: 2. Finished goods inventory, as of January 1, 20B, is 96,000 units. 3. Work-in-process inventory will remain constant. 4. Actual annual sales in 20A, including the estimate for December, were 350,000 units. 5. The average finished goods inventory during 20A was 70,000 units. IT IS REQUESTED. 1. Prepare the annual production budget, assuming that management policy is to budget ending finished goods inventory at a standard quantity, based on the ratio of historical sales of 20A to inventory turnover. 2. Prepare a schedule showing sales, production, and inventory levels for each month, assuming: 1) stable inventory, 2) stable production, and 3) recommended inventory-production levels. In developing your recommendations, assume that the following policies have been established: a) The president has set the policy that a maximum inventory of 85,000 units and a minimum inventory of 75,000 units should be used, except in abnormal circumstances. b) A stable level of production is definitely preferred, except that during the holiday season in July and August, production may be reduced by 25 percent. Likewise, a variation in production of 7.5 percent above and below the average level is acceptable. 3. What are the main problems faced by the company in production planning? Make your general recommendations.
3(2+x)-2(2x+6)=20-4x
a) A tap can supply eight gallons of gasoline daily to each of its 250 customers for 60 days. By how many gallons should each customer's daily supply be reduced so that it can supply 50 more customers for twenty more days?
What’s 20% of 125?
(5u + 6)-(3u+2)=
Solve this mathematical problem if 3/5 of a roll of tape measures 2m. How long is the complete roll?
A merchant can sell 20 electric shavers a day at a price of 25 each, but he can sell 30 if he sets a price of 20 for each electric shaver. Determine the demand equation, assuming it is linear. Consider (P= price, X= quantity demanded)
In a order to compare the means of two populations, independent random samples of 410 observations are selected from each population, with Sample 1 the results found in the table to the right. Complete parts a through e below. X1 = 5,319 S1= 143 a. Use a 95% confidence interval to estimate the difference between the population means (H - H2) Interpret the contidence interval. The contidence interval IS (Round to one decimal place as needed.) Sample 2 X2 = 5,285 S2 = 198 Aa. Use a 95% confidence interval to estimate the difference between the population means (A1 - M2) Interpret the contidence interval. The contidence interval Is (Round to one decimal place as needed.) b. Test the null hypothesis Ho versus alternative hypothesis Ha (H What is the test statistic? H2) + Give the significance level of the test, and interpret the result. Use a = 0.05. Z=
In an audience of 4000 people, 2 people are chosen, at random, to appear on stage. How many ways can the people be chosen?
In a company dedicated to packaging beer in 750 mL containers, a normal distribution is handled in its packaging process, which registers an average of 745 mL and a standard deviation of 8 mL. Determine: a) The probability that a randomly selected container exceeds 765 mL of beer b) The probability that the beer content of a randomly selected container is between 735 and 755 mL.
9.25=2pi r solve for r
What is the value of f(-3) for the function X squared+5x-8=
Nancy is a waitress at Seventh Heaven Hamburgers. She wants to estimate the average amount each table leaves for a tip. A random sample of 5 groups was taken and the amount they left for a tip (in dollars) is listed below: $11.00 $8.00 $6.00 $3.00 $7.00 a.) Find a 90% confidence interval for the average amount left by all groups. (*round to the nearest cent*) $ < μ < $ b.) If the sample size were larger, with everything else remaining the same, would the margin of Error increase or decrease? Decrease Increase c.) If the Confidence level were 95% instead of 90%, would the range (size) of the Confidence Interval be larger or smaller? Larger Smaller
solid obtained by rotation around the axis x = -1, the region delimited by x^2 - x + y = 0 and the abscissa axis
In an economy with C= 10+0.8 Yd ; I= 20+0.1Y ; G= 100 ; X= 20 ; M=10+0.2Y ; T=-10+0.2Y and R= 10, when knew that Yd= Y-T+R. How much is the budget? A. -23.18 B. -28.13 C. -13.28 D. -32.18
If the mean of the following numbers is 17, find the c value. Produce an algebraic solution. Guess and check is unacceptable. 12, 18, 21, c, 13
What js the greatest 4-digit even number that can be formed by 3,6,1,4?
Slope (7,3) and (9,5)
Write an equation of the affine function whose graph is perpendicular to the graph of f(x) = 5x − 1 and passes through the point (5, 20).