3.5E: Exercises for Section 3.5
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- Jun 30, 2021
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In exercises 1 - 10, find dydx for the given functions.
1) y=x2−secx+1
- Answer
- dydx=2x−secxtanx
2) y=3cscx+5x
3) y=x2cotx
- Answer
- dydx=2xcotx−x2csc2x
4) y=x−x3sinx
5) y=secxx
- Answer
- dydx=xsecxtanx−secxx2
6) y=sinxtanx
7) y=(x+cosx)(1−sinx)
- Answer
- dydx=(1−sinx)(1−sinx)−cosx(x+cosx)
8) y=tanx1−secx
9) y=1−cotx1+cotx
- Answer
- dydx=2csc2x(1+cotx)2
10) y=(cosx)(1+cscx)
In exercises 11 - 16, find the equation of the tangent line to each of the given functions at the indicated values of x. Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct.
11) [T] f(x)=−sinx,x=0
- Answer
-
y=−x
12) [T] f(x)=cscx,x=π2
13) [T] f(x)=1+cosx,x=3π2
- Answer
-
y=x+2−3π2
14) [T] f(x)=secx,x=π4
15) [T] f(x)=x2−tanx,x=0
- Answer
-
y=−x
16) [T] f(x)=5cotx,x=π4
In exercises 17 - 22, find d2ydx2 for the given functions.
17) y=xsinx−cosx
- Answer
- d2ydx2=3cosx−xsinx
18) y=sinxcosx
19) y=x−12sinx
- Answer
- d2ydx2=12sinx
20) y=1x+tanx
21) y=2cscx
- Answer
- d2ydx2=2csc(x)(csc2(x)+cot2(x))
22) y=sec2x
23) Find all x values on the graph of f(x)=−3sinxcosx where the tangent line is horizontal.
- Answer
- x=(2n+1)π4,where n is an integer
24) Find all x values on the graph of f(x)=x−2cosx for 0<x<2π where the tangent line has slope 2.
25) Let f(x)=cotx. Determine the points on the graph of f for 0<x<2π where the tangent line(s) is (are) parallel to the line y=−2x.
- Answer
- (π4,1),(3π4,−1),(5π4,1),(7π4,−1)
26) [T] A mass on a spring bounces up and down in simple harmonic motion, modeled by the function s(t)=−6cost where s is measured in inches and t is measured in seconds. Find the rate at which the spring is oscillating at t=5 s.
27) Let the position of a swinging pendulum in simple harmonic motion be given by s(t)=acost+bsint. Find the constants a and b such that when the velocity is 3 cm/s, s=0 and t=0.
- Answer
- a=0,b=3
28) After a diver jumps off a diving board, the edge of the board oscillates with position given by s(t)=−5cost cm at t seconds after the jump.
a. Sketch one period of the position function for t≥0.
b. Find the velocity function.
c. Sketch one period of the velocity function for t≥0.
d. Determine the times when the velocity is 0 over one period.
e. Find the acceleration function.
f. Sketch one period of the acceleration function for t≥0.
29) The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by y=10+5sinx where y is the number of hamburgers sold and x represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find y′ and determine the intervals where the number of burgers being sold is increasing.
- Answer
- y′=5cos(x), increasing on (0,π2),(3π2,5π2), and (7π2,12)
30) [T] The amount of rainfall per month in Phoenix, Arizona, can be approximated by y(t)=0.5+0.3cost, where t is months since January. Find y′and use a calculator to determine the intervals where the amount of rain falling is decreasing.
For exercises 31 - 33, use the quotient rule to derive the given equations.
31) ddx(cotx)=−csc2x
32) ddx(secx)=secxtanx
33) ddx(cscx)=−cscxcotx
34) Use the definition of derivative and the identity cos(x+h)=cosxcosh−sinxsinh to prove that ddx(cosx)=−sinx.
For exercises 35 - 39, find the requested higher-order derivative for the given functions.
35) d3ydx3 of y=3cosx
- Answer
- d3ydx3=3sinx
36) d2ydx2 of y=3sinx+x2cosx
37) d4ydx4 of y=5cosx
- Answer
- d4ydx4=5cosx
38) d2ydx2 of y=secx+cotx
39) d3ydx3 of y=x10−secx
- Answer
- d3ydx3=720x7−5tan(x)sec3(x)−tan3(x)sec(x)