Section 3.1 Enhanced Assignment

Instructor: Neelam Aggarwal Course: Spring 2022 - Calculus I (MATH1441-02F) Student: Jalen Wilson Date: 07/12/24 Assignment: Section 3.1 Enhanced Assignment 2 1. Find an equation for the line tangent to y = − 4 − 2x at ​( − 3​, − 22​). 2 The equation for the line tangent to y = − 4 − 2x at ​( − 3​, − 22​) is y = 12x + 14 . 2. Find an equation for the tangent to the curve at the given point. Then sketch the curve and the tangent together. 3 y = x ​, ​(2​,8​) y= 12x − 16 Choose the correct graph of the curve and the tangent below. A. B. 20 C. y 20 10 y 20 10 10 x -4 -2 2 4 y x -4 -2 2 x 4 -4 -2 2 -10 -10 -10 -20 -20 -20 4 3. Find the slope of the​function's graph at the given point. Then find an equation for the line tangent to the graph there. 2 f(x) = x + 1​, ​(2​,5​) What is the slope of the​function's graph at the given​point? m= 4 ​(Simplify your answer.) Find an equation for the line tangent to the graph at the given point. y= 4. 4x − 3 Find the slope of the graph of the function ​g(x) = The slope of the graph of the function ​g(x) = 6x x+5 6x x+5 at ​( − 3​, − 9​). Then find an equation for the line tangent to the graph at that point. 15 at ​( − 3​, − 9​) is . 2 ​(Type an integer or a simplified​fraction.) The equation for the line tangent to ​g(x) = 6x x+5 at ​( − 3​, − 9​) is y = 15 2 x+ 27 2 . 5. Find the slope of the​function's graph at the given point. Then find an equation for the line tangent to the graph there. 3 h(t) = − 2t ​, ​(1​, − 2​) What is the slope of the​function's graph at the given​point? m= −6 ​(Simplify your​answer.) Find an equation for the line tangent to the graph at the given point. y= − 6t + 4 6. Find the slope of the graph of the function ​f(x) = The slope of the graph of the function ​f(x) = 2x at ​(8​,4​). Then find an equation of the line tangent to the graph at that point. 1 2x at ​(8​,4​) is . 4 ​(Type an integer or a simplified​fraction.) The equation of the line tangent to the graph of ​f(x) = 1 2x at ​(8​,4​) is y = 4 x+2 . 7. Find the slope of the curve y = − 4x2 at ​(2​, − 16​). 2 The slope of the curve y = − 4x at ​(2​, − 16​) is − 16 . ​(Simplify your​answer.) 8. Find the slope of the following curve at x = 8. y= 1 x−5 The slope of the given curve at x = 8 is − 1 9 . ​(Simplify your​answer.) 9. Find the point​(x,y), at which the graph of y = 6x2 + 7x − 3 has a horizontal tangent. 2 The function y = 6x + 7x − 3 has a horizontal tangent at − 7 12 ,− 121 24 . ​(Type an ordered pair. Type simplified​fractions.) 10. Find equations of all lines having slope − 2 that are tangent to the curve y = 8 x−7 . Select the correct choice below and fill in the answer​box(es) within your choice. A. There is only one line tangent to the curve with a slope of − 2 and its equation is . ​(Type an​equation.) B. There are two lines tangent to the curve with a slope of − 2. The equation of the line with the larger​y-intercept is y = − 2x + 22 and the equation of the line with the smaller​y-intercept is y = − 2x + 6 . ​(Type equations.) 2 11. An object is dropped from the top of a cliff 600 meters high. Its height above the ground t seconds after it is dropped is 600 − 4.9t . Determine its speed 5 seconds after it is dropped. The speed of the object 5 seconds after it is dropped is ​(Simplify your​answer.) 49 ​m/sec. 12. At t sec after​liftoff, the height of a rocket is 2 ft. How fast is the rocket climbing sec after the​liftoff? 2t 8 The speed of the rocket 8 seconds after liftoff is ​(Simplify your​answer.) 32 2 13. What is the rate of change of the area of a square A = s The area changes at a rate of 24 ​(Type an exact​answer, using π as​needed.) . ​ft/sec. with respect to the side length when the side length is s = 12​? 14. Calculate the derivative of the function. Then find the value of the derivative as specified. 2 ​g(x) = 3x − ​4x; g′​(3) A. g′​(x) = ​6x; g′​(3) = 18 B. g′​(x) = 6x − ​4; g′​(3) = 14 C. g′​(x) = 2x − ​4; g′​(3) = 2 D. g′​(x) = 3x − ​4; g′​(3) = 5 15. Find the indicated derivative. dv dt if v = t + A. B. 1− 5 t 5 1− t C. t− 5 t D. 1+ 2 2 5 t 2 5 t